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1. Uniform Stability of a Particle Approximation of the Optimal Filter Derivative

Author

Arnaud Doucet, Sumeetpal S. Singh, Pierre Del Moral, Department of Statistics [Oxford], University of Oxford [Oxford], Statistical Laboratory [Cambridge], Department of Pure Mathematics and Mathematical Statistics (DPMMS), Faculty of mathematics Centre for Mathematical Sciences [Cambridge] (CMS), University of Cambridge [UK] (CAM)-University of Cambridge [UK] (CAM)-Faculty of mathematics Centre for Mathematical Sciences [Cambridge] (CMS), University of Cambridge [UK] (CAM)-University of Cambridge [UK] (CAM), Advanced Learning Evolutionary Algorithms (ALEA), Inria Bordeaux - Sud-Ouest, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Université de Bordeaux (UB)-Centre National de la Recherche Scientifique (CNRS), Institut de Mathématiques de Bordeaux (IMB), 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), Dept of Statistics & Dept of Computer Science, University of British Columbia (UBC), University of Cambridge [UK] (CAM), Inria & Labri, Univ. Bordeaux, Apollo - University of Cambridge Repository, and Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1 (UB)-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)

Subjects
FOS: Computer and information sciences, Control and Optimization, Mathematics - Statistics Theory, Statistics Theory (math.ST), Derivative, sequential Monte Carlo, state-space models, Statistics - Computation, Stability (probability), Methodology (stat.ME), FOS: Mathematics, recursive maximum likelihood, Applied mathematics, Computation (stat.CO), Statistics - Methodology, ComputingMilieux_MISCELLANEOUS, Mathematics, [STAT.AP]Statistics [stat]/Applications [stat.AP], Signal processing, hidden Markov models, Estimation theory, Applied Mathematics, smoothing, Filter (signal processing), Statistics::Computation, [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], Nonlinear system, filter derivative, Particle filter, Smoothing
Abstract

Sequential Monte Carlo methods, also known as particle methods, are a widely used set of computational tools for inference in non-linear non-Gaussian state-space models. In many applications it may be necessary to compute the sensitivity, or derivative, of the optimal filter with respect to the static parameters of the state-space model; for instance, in order to obtain maximum likelihood model parameters of interest, or to compute the optimal controller in an optimal control problem. In Poyiadjis et al. [2011] an original particle algorithm to compute the filter derivative was proposed and it was shown using numerical examples that the particle estimate was numerically stable in the sense that it did not deteriorate over time. In this paper we substantiate this claim with a detailed theoretical study. Lp bounds and a central limit theorem for this particle approximation of the filter derivative are presented. It is further shown that under mixing conditions these Lp bounds and the asymptotic variance characterized by the central limit theorem are uniformly bounded with respect to the time index. We demon- strate the performance predicted by theory with several numerical examples. We also use the particle approximation of the filter derivative to perform online maximum likelihood parameter estimation for a stochastic volatility model.

Published
2015

2. Diffusivity of a random walk on random walks

Author

Serge Cohen, Thibault Espinasse, James Norris, Emmanuel Boissard, 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), Statistical Laboratory [Cambridge], Department of Pure Mathematics and Mathematical Statistics (DPMMS), Faculty of mathematics Centre for Mathematical Sciences [Cambridge] (CMS), University of Cambridge [UK] (CAM)-University of Cambridge [UK] (CAM)-Faculty of mathematics Centre for Mathematical Sciences [Cambridge] (CMS), University of Cambridge [UK] (CAM)-University of Cambridge [UK] (CAM), 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
Random graph, Discrete mathematics, Heterogeneous random walk in one dimension, Applied Mathematics, General Mathematics, Probability (math.PR), Central limit theorem, Loop-erased random walk, Random element, Random walk, 05C81, 60F05, Computer Graphics and Computer-Aided Design, Graph, [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], Combinatorics, Mathematics::Probability, Lévy flight, FOS: Mathematics, Quantum walk, Mathematics - Probability, ComputingMilieux_MISCELLANEOUS, Software, Self-avoiding walk, Mathematics
Abstract

We consider a random walk Zn1,',ZnK+1∈i¾?K+1 with the constraint that each coordinate of the walk is at distance one from the following one. In this paper, we show that this random walk is slowed down by a variance factor i¾?K2=2K+2 with respect to the case of the classical simple random walk without constraint. © 2014 Wiley Periodicals, Inc. Random Struct. Alg., 47, 267-283, 2015

Published
2014

3. Nonexplosion criteria for relativistic diffusions

Author

Jacques Franchi, Ismael Bailleul, Statistical Laboratory [Cambridge], Department of Pure Mathematics and Mathematical Statistics (DPMMS), Faculty of mathematics Centre for Mathematical Sciences [Cambridge] (CMS), University of Cambridge [UK] (CAM)-University of Cambridge [UK] (CAM)-Faculty of mathematics Centre for Mathematical Sciences [Cambridge] (CMS), University of Cambridge [UK] (CAM)-University of Cambridge [UK] (CAM), Institut de Recherche Mathématique Avancée (IRMA), Université de Strasbourg (UNISTRA)-Centre National de la Recherche Scientifique (CNRS), Agence Nationale de la Recherche, n° ANR-09-BLAN-0364-01, et EPSRC grant EP/E01772X/1., I. Bailleul - J. Franchi, and ProbaGeo, ANR-09-BLAN-0364-01,ProbaGeo, ANR-09-BLAN-0364-01

Subjects
Statistics and Probability, 83C99, General relativity, 83C10, Lorentz transformation, FOS: Physical sciences, nonexplosion criterion, Curvature, 01 natural sciences, Volume growth, 010104 statistics & probability, symbols.namesake, Mathematics::Probability, FOS: Mathematics, Covariant transformation, 0101 mathematics, Mathematical Physics, Mathematical physics, Mathematics, Primary 58J65, secondary 53C50, 83C10, 83C99, Probability (math.PR), 53C50, 010102 general mathematics, Subriemannian minimal time, Mathematical Physics (math-ph), 58J65, Physics::History of Physics, Manifold, Non-explosion criterion, [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], Relativistic diffusions, symbols, Statistics, Probability and Uncertainty, Mathematics - Probability, Stochastic completeness
Abstract

Some general Lorentz covariant operators, associated to the so-called \Theta (or \Xi)-relativistic diffusions and making sense in any Lorentzian manifold, have been introduced by Franchi and Le Jan [Comm. Pure Appl. Math. 60 (2007) 187-251], Franchi and Le Jan [Curvature diffusions in general relativity (2010). Unpublished manuscript]. Only a few examples have been studied so far. We provide in this work some nonexplosion criteria for these diffusions, which can be used in generic cases., Comment: Published in at http://dx.doi.org/10.1214/11-AOP672 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

Published
2012

4. A Historical Law of Large Numbers for the Marcus-Lushnikov Process

Author

Stéphanie Jacquot, Statistical Laboratory [Cambridge], Department of Pure Mathematics and Mathematical Statistics (DPMMS), Faculty of mathematics Centre for Mathematical Sciences [Cambridge] (CMS), University of Cambridge [UK] (CAM)-University of Cambridge [UK] (CAM)-Faculty of mathematics Centre for Mathematical Sciences [Cambridge] (CMS), University of Cambridge [UK] (CAM)-University of Cambridge [UK] (CAM), INRIA Bordeaux - Sud-Ouest, and SESSION 03 : Processus de branchement en dynamique des populations

Subjects
Statistics and Probability, 60B10, Distribution (number theory), Smoluchowski coagulation equation, tightness, 01 natural sciences, Combinatorics, 010104 statistics & probability, symbols.namesake, 60J25, Law of large numbers, 60B05, Statistical physics, Limit (mathematics), coupling, Physics::Chemical Physics, 0101 mathematics, Marcus-Lushnikov process on trees, Mathematics, Particle system, limit measure on trees, 010102 general mathematics, Empirical distribution function, [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], symbols, Particle, historical trees, 60F15, Tree (set theory), Statistics, Probability and Uncertainty
Abstract

The Marcus-Lushnikov process is a finite stochastic particle system, in which each particle is entirely characterized by its mass. Each pair of particles with masses $x$ and $y$ merges into a single particle at a given rate $K(x,y)$. Under certain assumptions, this process converges to the solution to the Smoluchowski coagulation equation, as the number of particles increases to infinity. The Marcus-Lushnikov process gives at each time the distribution of masses of the particles present in the system, but does not retain the history of formation of the particles. In this paper, we set up a historical analogue of the Marcus-Lushnikov process (built according to the rules of construction of the usual Markov-Lushnikov process) each time giving what we call the historical tree of a particle. The historical tree of a particle present in the Marcus-Lushnikov process at a given time t encodes information about the times and masses of the coagulation events that have formed that particle. We prove a law of large numbers for the empirical distribution of such historical trees. The limit is a natural measure on trees which is constructed from a solution to the Smoluchowski coagulation equation.

Published
2010

5. Almost All F-Free Graphs Have The Erdos-Hajnal Property

Author

Bruce Reed, Stéphan Thomassé, Martin Loebl, Alex Scott, Andrew Thomason, Department of Applied Mathematics (KAM) (KAM), Univerzita Karlova v Praze, Mathematical Institute [Oxford] (MI), University of Oxford, Statistical Laboratory [Cambridge], Department of Pure Mathematics and Mathematical Statistics (DPMMS), Faculty of mathematics Centre for Mathematical Sciences [Cambridge] (CMS), University of Cambridge [UK] (CAM)-University of Cambridge [UK] (CAM)-Faculty of mathematics Centre for Mathematical Sciences [Cambridge] (CMS), University of Cambridge [UK] (CAM)-University of Cambridge [UK] (CAM), Laboratoire de l'Informatique du Parallélisme (LIP), École normale supérieure de Lyon (ENS de Lyon)-Université Claude Bernard Lyon 1 (UCBL), Université de Lyon-Université de Lyon-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre National de la Recherche Scientifique (CNRS), Algorithmes, Graphes et Combinatoire (ALGCO), Laboratoire d'Informatique de Robotique et de Microélectronique de Montpellier (LIRMM), Université de Montpellier (UM)-Centre National de la Recherche Scientifique (CNRS)-Université de Montpellier (UM)-Centre National de la Recherche Scientifique (CNRS), University of Oxford [Oxford], Centre National de la Recherche Scientifique (CNRS)-Université de Lyon-Institut National de Recherche en Informatique et en Automatique (Inria)-Université Claude Bernard Lyon 1 (UCBL), Université de Lyon-École normale supérieure - Lyon (ENS Lyon), Centre National de la Recherche Scientifique (CNRS)-Université de Montpellier (UM)-Centre National de la Recherche Scientifique (CNRS)-Université de Montpellier (UM), and École normale supérieure - Lyon (ENS Lyon)-Université Claude Bernard Lyon 1 (UCBL)

Subjects
Discrete mathematics, 010102 general mathematics, Induced subgraph, 0102 computer and information sciences, [INFO.INFO-DM]Computer Science [cs]/Discrete Mathematics [cs.DM], 16. Peace & justice, 01 natural sciences, Degeneracy (graph theory), Combinatorics, Rado graph, 010201 computation theory & mathematics, Graph power, Triangle-free graph, Split graph, 0101 mathematics, Forbidden graph characterization, Mathematics, Universal graph
Abstract

International audience; Erdős and Hajnal conjectured that, for every graph H, there exists a constant ɛ(H) > 0 such that every H-free graph G (that is, not containing H as an induced subgraph) must contain a clique or an independent set of size at least |G|ɛ( H). We prove that there exists ɛ(H) such that almost every H-ïvee graph G has this property, meaning that, amongst the if-free graphs with n vertices, the proportion having the property tends to one as n → ∞.

Published
2010

6. A stochastic min-driven coalescence process and its hydrodynamical limit

Author

Anne-Laure Basdevant, Clément Rau, Philippe Laurençot, James Norris, Modélisation aléatoire de Paris X (MODAL'X), Université Paris Nanterre (UPN), 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), Statistical Laboratory [Cambridge], Department of Pure Mathematics and Mathematical Statistics (DPMMS), Faculty of mathematics Centre for Mathematical Sciences [Cambridge] (CMS), University of Cambridge [UK] (CAM)-University of Cambridge [UK] (CAM)-Faculty of mathematics Centre for Mathematical Sciences [Cambridge] (CMS), University of Cambridge [UK] (CAM)-University of Cambridge [UK] (CAM), 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), Université Toulouse 1 Capitole (UT1), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), and Institut National des Sciences Appliquées (INSA)-Université Fédérale Toulouse Midi-Pyrénées-Institut National des Sciences Appliquées (INSA)-Université Toulouse - Jean Jaurès (UT2J)-Université Toulouse III - Paul Sabatier (UT3)

Subjects
Statistics and Probability, Stochastic modelling, stochastic coalescence, 82C22, 60K35, 60H10, 34A34, 34C11, Binary number, 01 natural sciences, hydrodynamical limit, 010104 statistics & probability, 82C22, 60K35, 60H10, 34A34, 34C11, FOS: Mathematics, Statistical physics, 0101 mathematics, Mathematics, Coalescence (physics), Probability (math.PR), 010102 general mathematics, Time evolution, [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], min-driven clustering, Statistics, Probability and Uncertainty, Mathematics - Probability
Abstract

International audience; A stochastic system of particles is considered in which the sizes of the particles increase by successive binary mergers with the constraint that each coagulation event involves a particle with minimal size. Convergence of a suitably renormalised version of this process to a deterministic hydrodynamical limit is shown and the time evolution of the minimal size is studied for both deterministic and stochastic models.

Published
2009

7. Large deviations for the Yang-Mills measure on a compact surface

Author

James Norris, Thierry Lévy, Département de Mathématiques et Applications - ENS Paris (DMA), École normale supérieure - Paris (ENS Paris), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS), Statistical Laboratory [Cambridge], Department of Pure Mathematics and Mathematical Statistics (DPMMS), Faculty of mathematics Centre for Mathematical Sciences [Cambridge] (CMS), University of Cambridge [UK] (CAM)-University of Cambridge [UK] (CAM)-Faculty of mathematics Centre for Mathematical Sciences [Cambridge] (CMS), University of Cambridge [UK] (CAM)-University of Cambridge [UK] (CAM), Centre National de la Recherche Scientifique (CNRS)-École normale supérieure - Paris (ENS Paris), and Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)

Subjects
Surface (mathematics), Mathematics - Differential Geometry, High Energy Physics::Lattice, FOS: Physical sciences, Yang–Mills existence and mass gap, 01 natural sciences, Measure (mathematics), General Relativity and Quantum Cosmology, High Energy Physics::Theory, Large deviation principle, Simple (abstract algebra), [MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph], Mathematics::Quantum Algebra, 0103 physical sciences, FOS: Mathematics, Limit (mathematics), 0101 mathematics, Sobolev connections, Yang-Mills measure, Mathematical Physics, Mathematics, MSC 60F10, 81T13, 58D20, 010102 general mathematics, Mathematical analysis, Probability (math.PR), Statistical and Nonlinear Physics, Mathematical Physics (math-ph), [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], Nonlinear Sciences::Exactly Solvable and Integrable Systems, Differential Geometry (math.DG), [MATH.MATH-DG]Mathematics [math]/Differential Geometry [math.DG], Large deviations theory, 010307 mathematical physics, Rate function, Energy (signal processing), Mathematics - Probability
Abstract

We prove the first mathematical result relating the Yang-Mills measure on a compact surface and the Yang-Mills energy. We show that, at the small volume limit, the Yang-Mills measures satisfy a large deviation principle with a rate function which is expressed in a simple and natural way in terms of the Yang-Mills energy.

Published
2004
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8. A finite dimensional approach to Donaldson’s J‑flow

Author

Ruadhaí Dervan, Julien Keller, Department of Pure Mathematics and Mathematical Statistics (DPMMS), Faculty of mathematics Centre for Mathematical Sciences [Cambridge] (CMS), University of Cambridge [UK] (CAM)-University of Cambridge [UK] (CAM), Institut de Mathématiques de Marseille (I2M), Aix Marseille Université (AMU)-École Centrale de Marseille (ECM)-Centre National de la Recherche Scientifique (CNRS), ANR-14-CE25-0010,EMARKS,Métriques extrémales et K-stabilité(2014), ANR-11-IDEX-0001,Amidex,INITIATIVE D'EXCELLENCE AIX MARSEILLE UNIVERSITE(2011), and Apollo - University of Cambridge Repository

Subjects
Mathematics - Differential Geometry, Statistics and Probability, Pure mathematics, Stability (learning theory), polarisation, Type (model theory), Mathematics - Algebraic Geometry, J-flow, Mathematics::Algebraic Geometry, balanced metrics, FOS: Mathematics, Uniqueness, Complex Variables (math.CV), Algebraic Geometry (math.AG), Mathematics, Energy functional, critical metrics, Mathematics - Complex Variables, Linear system, 4904 Pure Mathematics, K-stability, [MATH.MATH-CV]Mathematics [math]/Complex Variables [math.CV], Chow stability, canonical metrics, Canonical bundle, Differential Geometry (math.DG), projective manifold, Flow (mathematics), [MATH.MATH-DG]Mathematics [math]/Differential Geometry [math.DG], 49 Mathematical Sciences, 14L24, 53C07, 32Q20, 53D50, 53D20, quantization, [MATH.MATH-AG]Mathematics [math]/Algebraic Geometry [math.AG], Geometry and Topology, Statistics, Probability and Uncertainty, Complex manifold, Analysis
Abstract

We define a quantisation of the J-flow over a projective complex manifold. As corollaries, we obtain new proofs of uniqueness of critical points of the J-flow and that these critical points achieve the absolute minimum of an associated energy functional. We show that the existence of a critical point of the J-flow implies the existence of certain canonical metrics, that we call J-balanced metrics. We define a notion of Chow stability for linear systems and relate it to the existence of J-balanced metrics. We also relate the asymptotic Chow stability of a linear system to an analogue of K-semistability that was introduced by Lejmi-Sz\'ekelyhidi, which we call J-semistability. Then, we relate J-semistability to K-stability when one of the polarisation is the canonical bundle. Eventually, this gives new K-stable polarisations of surfaces of general type., Comment: 47 pages; Revised version. To appear in Comm. Anal. Geom

Published
2019

9. From Boltzmann to incompressible Navier–Stokes in Sobolev spaces with polynomial weight

Author

Sara Merino-Aceituno, Marc Briant, Clément Mouhot, 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), DPMMS/CMS, University of Cambridge [UK] (CAM), Department of Pure Mathematics and Mathematical Statistics (DPMMS), Faculty of mathematics Centre for Mathematical Sciences [Cambridge] (CMS), and University of Cambridge [UK] (CAM)-University of Cambridge [UK] (CAM)

Subjects
Polynomial, Boltzmann equation on the Torus, Mathematics::Analysis of PDEs, FOS: Physical sciences, 010103 numerical & computational mathematics, Space (mathematics), Lebesgue integration, Incompressible Navier-Stokes hydrodynamical limit, 01 natural sciences, Physics::Fluid Dynamics, symbols.namesake, 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], QA276, 0101 mathematics, Exponential decay, Mathematical Physics, Mathematics, Applied Mathematics, 010102 general mathematics, Mathematical analysis, Mathematical Physics (math-ph), 16. Peace & justice, Boltzmann equation, Sobolev space, Boltzmann constant, symbols, Knudsen number, Cauchy theory, Analysis, Analysis of PDEs (math.AP)
Abstract

We study the Boltzmann equation on the $d$-dimensional torus in a perturbative setting around a global equilibrium under the Navier-Stokes linearisation. We use a recent functional analysis breakthrough to prove that the linear part of the equation generates a $C^0$-semigroup with exponential decay in Lebesgue and Sobolev spaces with polynomial weight, independently on the Knudsen number. Finally we show a Cauchy theory and an exponential decay for the perturbed Boltzmann equation, uniformly in the Knudsen number, in Sobolev spaces with polynomial weight. The polynomial weight is almost optimal and furthermore, this result only requires derivatives in the space variable and allows to connect to solutions to the incompressible Navier-Stokes equations in these spaces., 30 pages

Published
2018

10. On blow up for the energy super critical defocusing nonlinear Schrödinger equations

Author

Jeremie Szeftel, Igor Rodnianski, Pierre Raphaël, Frank Merle, Analyse, Géométrie et Modélisation (AGM - UMR 8088), CY Cergy Paris Université (CY)-Centre National de la Recherche Scientifique (CNRS), Department of Pure Mathematics and Mathematical Statistics (DPMMS), Faculty of mathematics Centre for Mathematical Sciences [Cambridge] (CMS), University of Cambridge [UK] (CAM)-University of Cambridge [UK] (CAM), Department of Mathematics - Princeton University, Princeton University, Laboratoire Jacques-Louis Lions (LJLL (UMR_7598)), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université de Paris (UP), and Apollo - University of Cambridge Repository

Subjects
4902 Mathematical Physics, General Mathematics, 010102 general mathematics, Dimension (graph theory), Mathematics::Analysis of PDEs, 4904 Pure Mathematics, MSC: 35Q55, 01 natural sciences, Schrödinger equation, Euler equations, symbols.namesake, Singularity, 0103 physical sciences, symbols, 49 Mathematical Sciences, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], 010307 mathematical physics, Soliton, 0101 mathematics, Degeneracy (mathematics), Nonlinear Schrödinger equation, Nonlinear Sciences::Pattern Formation and Solitons, Energy (signal processing), Mathematics, Mathematical physics
Abstract

We consider the energy supercritical defocusing nonlinear Schrödinger equation $$\begin{aligned} i\partial _tu+\Delta u-u|u|^{p-1}=0 \end{aligned}$$ i ∂ t u + Δ u - u | u | p - 1 = 0 in dimension $$d\ge 5$$ d ≥ 5 . In a suitable range of energy supercritical parameters (d, p), we prove the existence of $${\mathcal {C}}^\infty $$ C ∞ well localized spherically symmetric initial data such that the corresponding unique strong solution blows up in finite time. Unlike other known blow up mechanisms, the singularity formation does not occur by concentration of a soliton or through a self similar solution, which are unknown in the defocusing case, but via a front mechanism. Blow up is achieved by compression for the associated hydrodynamical flow which in turn produces a highly oscillatory singularity. The front blow up profile is chosen among the countable family of $${\mathcal {C}}^\infty $$ C ∞ spherically symmetric self similar solutions to the compressible Euler equation whose existence and properties in a suitable range of parameters are established in the companion paper (Merle et al. in Preprint (2019)) under a non degeneracy condition which is checked numerically.

Published
2021

11. Strong Ramsey Games in Unbounded Time

Author

Ivailo Hartarsky, Marius Tiba, Stefan David, Department of Pure Mathematics and Mathematical Statistics (DPMMS), Faculty of mathematics Centre for Mathematical Sciences [Cambridge] (CMS), University of Cambridge [UK] (CAM)-University of Cambridge [UK] (CAM), Département de Mathématiques et Applications - ENS Paris (DMA), École normale supérieure - Paris (ENS Paris), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS), Centre National de la Recherche Scientifique (CNRS)-École normale supérieure - Paris (ENS Paris), and Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)

Subjects
05C57 (Primary) 91A43, 91A46 (Secondary), Hypergraph, Computer Science::Computer Science and Game Theory, Strong games, 010102 general mathematics, 0102 computer and information sciences, 01 natural sciences, Graph, 05C57, 91A43, 91A46, Combinatorics, Compact space, Ramsey games, 010201 computation theory & mathematics, Bounded function, [MATH.MATH-CO]Mathematics [math]/Combinatorics [math.CO], FOS: Mathematics, Mathematics - Combinatorics, Discrete Mathematics and Combinatorics, Combinatorics (math.CO), 0101 mathematics, Mathematics, Positional games
Abstract

For two graphs $B$ and $H$ the strong Ramsey game $\mathcal{R}(B,H)$ on the board $B$ and with target $H$ is played as follows. Two players alternately claim edges of $B$. The first player to build a copy of $H$ wins. If none of the players win, the game is declared a draw. A notorious open question of Beck asks whether the first player has a winning strategy in $\mathcal{R}(K_n,K_k)$ in bounded time as $n\rightarrow\infty$. Surprisingly, in a recent paper Hefetz et al. constructed a $5$-uniform hypergraph $\mathcal{H}$ for which they proved that the first player does not have a winning strategy in $\mathcal{R}(K_n^{(5)},\mathcal{H})$ in bounded time. They naturally ask whether the same result holds for graphs. In this paper we make further progress in decreasing the rank. In our first result, we construct a graph $G$ (in fact $G=K_6\setminus K_4$) and prove that the first player does not have a winning strategy in $\mathcal{R}(K_n \sqcup K_n,G)$ in bounded time. As an application of this result we deduce our second result in which we construct a $4$-uniform hypergraph $G'$ and prove that the first player does not have a winning strategy in $\mathcal{R}(K_n^{(4)},G')$ in bounded time. This improves the result in the paper above. An equivalent formulation of our first result is that the game $\mathcal{R}(K_\omega\sqcup K_\omega,G)$ is a draw. Another reason for interest on the board $K_\omega\sqcup K_\omega$ is a folklore result that the disjoint union of two finite positional games both of which are first player wins is also a first player win. An amusing corollary of our first result is that at least one of the following two natural statements is false: (1) for every graph $H$, $\mathcal{R}(K_\omega,H)$ is a first player win; (2) for every graph $H$ if $\mathcal{R}(K_\omega,H)$ is a first player win, then $\mathcal{R}(K_\omega\sqcup K_\omega,H)$ is also a first player win., Comment: 17 pages, 48 figures; improved presentation, particularly in section 3

Published
2020

12. SALEM NUMBERS AND THE SPECTRUM OF HYPERBOLIC SURFACES

Author

Bertrand Deroin, Emmanuel Breuillard, Deroin, Bertrand, Analyse, Géométrie et Modélisation (AGM - UMR 8088), CY Cergy Paris Université (CY)-Centre National de la Recherche Scientifique (CNRS), Department of Pure Mathematics and Mathematical Statistics (DPMMS), Faculty of mathematics Centre for Mathematical Sciences [Cambridge] (CMS), and University of Cambridge [UK] (CAM)-University of Cambridge [UK] (CAM)

Subjects
Pure mathematics, Conjecture, Salem number, Mathematics - Number Theory, General Mathematics, 010102 general mathematics, Spectrum (functional analysis), Mathematics::Classical Analysis and ODEs, Cheeger constant, [MATH] Mathematics [math], 01 natural sciences, first eigenvalue of the Laplacian, Mathematics - Spectral Theory, arithmetic surfaces, 0103 physical sciences, FOS: Mathematics, Spectral gap, 010307 mathematical physics, Number Theory (math.NT), [MATH]Mathematics [math], 0101 mathematics, Spectral Theory (math.SP), Mathematics
Abstract

We give a reformulation of Salem's conjecture about the absence of Salem numbers near one in terms of a uniform spectral gap for certain arithmetic hyperbolic surfaces., Comment: 13 pages

Published
2020

13. Hypocoercivity without confinement

Author

Jean Dolbeault, Christian Schmeiser, Emeric Bouin, Clément Mouhot, Stéphane Mischler, CEntre de REcherches en MAthématiques de la DEcision (CEREMADE), Université Paris Dauphine-PSL, Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS), Department of Pure Mathematics and Mathematical Statistics (DPMMS), Faculty of mathematics Centre for Mathematical Sciences [Cambridge] (CMS), University of Cambridge [UK] (CAM)-University of Cambridge [UK] (CAM), Fakultät für Mathematik [Wien], Universität Wien, ANR-13-BS01-0004,KIBORD,Modèles cinétiques en biologie et domaines connexes(2013), ANR-12-BS01-0019,STAB,Stabilité du comportement asymptotique d'EDP, de processus stochastiques et de leurs discrétisations.(2012), ANR-17-CE40-0030,EFI,Entropie, flots, inégalités(2017), European Project: 279600,EC:FP7:ERC,ERC-2011-StG_20101014,MATKIT(2011), European Project: 726386,MAFRAN, Centre National de la Recherche Scientifique (CNRS)-Université Paris Dauphine-PSL, and Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)

Subjects
Fokker-Planck operator, scattering operator, Function space, factorization method, 82C40, Poincaré inequality, Ocean Engineering, 35H10, diffusion limit, Space (mathematics), 01 natural sciences, symbols.namesake, Hypocoercivity, Mathematics - Analysis of PDEs, micro/macro decomposition, 35Q84, FOS: Mathematics, Applied mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], 0101 mathematics, Exponential decay, transport operator, linear kinetic equations, Mathematics, 76P05, 35K65, 35P15, Dirichlet form, 010102 general mathematics, Nash's inequality, Green's function, Fourier mode decomposition, Exponential function, Fokker–Planck operator, 010101 applied mathematics, Moment (mathematics), symbols, Heat equation, micro-/macrodecomposition, Analysis of PDEs (math.AP)
Abstract

International audience; In this paper, hypocoercivity methods are applied to linear kinetic equations with mass conservation and without confinement, in order to prove that the solutions have an algebraic decay rate in the long-time range, which the same as the rate of the heat equation. Two alternative approaches are developed: an analysis based on decoupled Fourier modes and a direct approach where, instead of the Poincar\'e inequality for the Dirichlet form, Nash's inequality is employed. The first approach is also used to provide a simple proof of exponential decay to equilibrium on the flat torus. The results are obtained on a space with exponential weights and then extended to larger function spaces by a factorization method. The optimality of the rates is discussed. Algebraic rates of decay on the whole space are improved when the initial datum has moment cancellations.

Published
2020
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14. Learning optical flow for fast MRI reconstruction

Author

Veronica Corona, Carola-Bibiane Schönlieb, Angelica I. Aviles-Rivero, Noémie Debroux, T. Schmoderer, Laboratoire de Mathématiques de l'INSA de Rouen Normandie (LMI), Institut national des sciences appliquées Rouen Normandie (INSA Rouen Normandie), Institut National des Sciences Appliquées (INSA)-Normandie Université (NU)-Institut National des Sciences Appliquées (INSA)-Normandie Université (NU), Department of Pure Mathematics and Mathematical Statistics (DPMMS), Faculty of mathematics Centre for Mathematical Sciences [Cambridge] (CMS), University of Cambridge [UK] (CAM)-University of Cambridge [UK] (CAM), Department of Applied Mathematics and Theoretical Physics (DAMTP), Centrefor Mathematical Sciences,University of Cambridge, Institut Pascal (IP), Centre National de la Recherche Scientifique (CNRS)-Université Clermont Auvergne (UCA)-Institut national polytechnique Clermont Auvergne (INP Clermont Auvergne), Université Clermont Auvergne (UCA)-Université Clermont Auvergne (UCA), University of Cambridge [UK] (CAM), Dynamical Interconnected Systems in COmplex Environments (DISCO), Inria Saclay - Ile de France, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire des signaux et systèmes (L2S), CentraleSupélec-Université Paris-Saclay-Centre National de la Recherche Scientifique (CNRS)-CentraleSupélec-Université Paris-Saclay-Centre National de la Recherche Scientifique (CNRS), Laboratoire des signaux et systèmes (L2S), CentraleSupélec-Université Paris-Saclay-Centre National de la Recherche Scientifique (CNRS), Department of Applied Mathematics and Theoretical Physics (DAMTP), and SIGMA Clermont (SIGMA Clermont)-Université Clermont Auvergne [2017-2020] (UCA [2017-2020])-Centre National de la Recherche Scientifique (CNRS)

Subjects
MRI Reconstruction, ComputingMethodologies_IMAGEPROCESSINGANDCOMPUTERVISION, Optical flow, Iterative reconstruction, Imaging phantom, 030218 nuclear medicine & medical imaging, Theoretical Computer Science, 03 medical and health sciences, 0302 clinical medicine, Optical Flow, FOS: Mathematics, FOS: Electrical engineering, electronic engineering, information engineering, [INFO.INFO-IM]Computer Science [cs]/Medical Imaging, [INFO]Computer Science [cs], Mathematics - Numerical Analysis, Mathematical Physics, Mathematics, Motion compensation, Ground truth, Applied Mathematics, Dynamic data, Image and Video Processing (eess.IV), Numerical Analysis (math.NA), Electrical Engineering and Systems Science - Image and Video Processing, Dictionary Learning, Computer Science Applications, Compressed sensing, [INFO.INFO-TI]Computer Science [cs]/Image Processing [eess.IV], Signal Processing, Dynamic contrast-enhanced MRI, 94A08, 65K05, 68Q32, Multi-Task Model, [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], Algorithm, 030217 neurology & neurosurgery
Abstract

International audience; Reconstructing high-quality magnetic resonance images (MRIs) from undersampled raw data is of great interest from both technical and clinical point of views. To this date, however, it is still a mathematically and computationally challenging problem due to its severe ill-posedness, resulting from the highly undersampled data. Whilst a number of techniques have been presented to improve image reconstruction, they only account for spatio-temporal regularisation, which shows its limitations in several relevant scenarios including dynamic data. In this work, we propose a new mathematical model for the reconstruction of high-quality medical MRI from few measurements. Our proposed approach combines—in a multi-task and hybrid model—the traditional compressed sensing formulation for the reconstruction of dynamic MRI with motion compensation by learning an optical flow approximation. More precisely, we propose to encode the dynamics in the form of an optical flow model that is sparsely represented over a learned dictionary. This has the advantage that ground truth data is not required in the training of the optical flow term. Furthermore, we present an efficient optimisation scheme to tackle the non-convex problem based on an alternating splitting method. We demonstrate the potentials of our approach through an extensive set of numerical results using different datasets and acceleration factors. Our combined approach reaches and outperforms several state-of-the-art techniques for multi-tasking reconstruction and other classic variational reconstruction schemes. Finally, we show the ability of our technique to transfer phantom based knowledge to real datasets.

Published
2021

15. Quasineutral limit for Vlasov–Poisson via Wasserstein stability estimates in higher dimension

Author

Mikaela Iacobelli, Daniel Han-Kwan, Centre de Mathématiques Laurent Schwartz (CMLS), École polytechnique (X)-Centre National de la Recherche Scientifique (CNRS), Department of Pure Mathematics and Mathematical Statistics (DPMMS), Faculty of mathematics Centre for Mathematical Sciences [Cambridge] (CMS), and University of Cambridge [UK] (CAM)-University of Cambridge [UK] (CAM)

Subjects
Work (thermodynamics), Applied Mathematics, 010102 general mathematics, Mathematical analysis, Poisson distribution, 01 natural sciences, Stability (probability), 010101 applied mathematics, symbols.namesake, Dimension (vector space), Physics::Plasma Physics, Physics::Space Physics, symbols, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Limit (mathematics), 0101 mathematics, ComputingMilieux_MISCELLANEOUS, Analysis, Mathematics
Abstract

This work is concerned with the quasineutral limit of the Vlasov–Poisson system in two and three dimensions. We justify the formal limit for very small but rough perturbations of analytic initial data, generalizing the results of [12] to higher dimension.

Published
2017

16. Strong law of large numbers for the capacity of the Wiener sausage in dimension four

Author

Perla Sousi, Amine Asselah, Bruno Schapira, Apollo - University of Cambridge Repository, Laboratoire d'Analyse et de Mathématiques Appliquées (LAMA), Centre National de la Recherche Scientifique (CNRS)-Université Paris-Est Créteil Val-de-Marne - Paris 12 (UPEC UP12)-Fédération de Recherche Bézout-Université Paris-Est Marne-la-Vallée (UPEM), Institut de Mathématiques de Marseille (I2M), Aix Marseille Université (AMU)-École Centrale de Marseille (ECM)-Centre National de la Recherche Scientifique (CNRS), Department of Pure Mathematics and Mathematical Statistics (DPMMS), Faculty of mathematics Centre for Mathematical Sciences [Cambridge] (CMS), University of Cambridge [UK] (CAM)-University of Cambridge [UK] (CAM), ANR-16-CE93-0003,MALIN,Marches aléatoires en interaction(2016), Université Paris-Est Marne-la-Vallée (UPEM)-Fédération de Recherche Bézout-Université Paris-Est Créteil Val-de-Marne - Paris 12 (UPEC UP12)-Centre National de la Recherche Scientifique (CNRS), Centre National de la Recherche Scientifique (CNRS)-École Centrale de Marseille (ECM)-Aix Marseille Université (AMU), DPMMS/CMS, University of Cambridge [UK] (CAM), ANR-16-CE93-0003,MALIN,Marches aléatoires en interaction(2017), Schapira, Bruno, and Marches aléatoires en interaction - - MALIN2016 - ANR-16-CE93-0003 - AAPG2016 - VALID

Subjects
Statistics and Probability, [MATH.MATH-PR] Mathematics [math]/Probability [math.PR], Logarithm, Law of large numbers, Wiener sausage, Expected value, 01 natural sciences, and phrases Capacity, 010104 statistics & probability, symbols.namesake, Dimension (vector space), Intersection, Mathematics::Probability, FOS: Mathematics, Law of large, 0101 mathematics, Scaling, Mathematics, Capacity, 010102 general mathematics, Mathematical analysis, Probability (math.PR), 4901 Applied Mathematics, 16. Peace & justice, Green kernel, [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], 4905 Statistics, symbols, 49 Mathematical Sciences, Statistics, Probability and Uncertainty, Critical dimension, Analysis, Mathematics - Probability, 60F05, 60G50
Abstract

We prove a strong law of large numbers for the Newtonian capacity of a Wiener sausage in the critical dimension four., Comment: Substantially expanded: the asymptotic of the first moment is obtained, giving rise to a fully explicit law of large numbers

Published
2019

17. Gaussian lower bounds for the Boltzmann equation without cut-off

Author

Luis Silvestre, Clément Mouhot, Cyril Imbert, Département de Mathématiques et Applications - ENS Paris (DMA), École normale supérieure - Paris (ENS Paris)-Centre National de la Recherche Scientifique (CNRS), DPMMS/CMS, University of Cambridge [UK] (CAM), Department of Mathematics [Chicago], University of Chicago, European Project: 726386,MAFRAN, École normale supérieure - Paris (ENS-PSL), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS), Department of Pure Mathematics and Mathematical Statistics (DPMMS), Faculty of mathematics Centre for Mathematical Sciences [Cambridge] (CMS), University of Cambridge [UK] (CAM)-University of Cambridge [UK] (CAM), Centre National de la Recherche Scientifique (CNRS)-École normale supérieure - Paris (ENS Paris), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL), and École normale supérieure - Paris (ENS Paris)

Subjects
Gaussian, 01 natural sciences, Upper and lower bounds, Boltzmann equation, symbols.namesake, lower bounds, Maximum principle, Mathematics - Analysis of PDEs, Argument, FOS: Mathematics, Cutoff, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], 0101 mathematics, long-range interactions, Mathematics, conditional regularity, non cut-off, Applied Mathematics, Mathematical analysis, 010101 applied mathematics, Computational Mathematics, symbols, moments, Analysis, Analysis of PDEs (math.AP)
Abstract

The study of positivity of solutions to the Boltzmann equation goes back to Carleman (1933), and the initial argument of Carleman was developed byPulvirenti-Wennberg (1997), the second author and Briant (2015). The appearance of a lower bound with Gaussian decay had however remained an open question for long-range interactions (the so-called non-cutoff collision kernels). We answer this question and establish such Gaussian lower bound for solutions to the Boltzmann equation without cutoff, in the case of hard and moderately soft potentials, with spatial periodic conditions, and under the sole assumption that hydrodynamic quantities (local mass, local energy and local entropy density) remain bounded. The paper is mostly self-contained, apart from the uniform upper bound on the solution established by the third author (2016)., 14 pages

Published
2019

18. Orbital degeneracy loci and applications

Author

Fabio Tanturri, Sara Angela Filippini, Laurent Manivel, Vladimiro Benedetti, Institut de Mathématiques de Marseille (I2M), Aix Marseille Université (AMU)-École Centrale de Marseille (ECM)-Centre National de la Recherche Scientifique (CNRS), Department of Pure Mathematics and Mathematical Statistics (DPMMS), Faculty of mathematics Centre for Mathematical Sciences [Cambridge] (CMS), University of Cambridge [UK] (CAM)-University of Cambridge [UK] (CAM), This work has been carried out in the framework of the Labex Archim`ede (ANR-11-LABX-0033) and of the A*MIDEX project (ANR-11-IDEX-0001-02), fundedby the 'Investissements d’Avenir' French Government programme managed bythe French National Research Agency. The second author was also partially sup-ported by the Engineering and Physical Sciences Research Council programmegrant 'Classification, Computation, and Construction: New Methods in Geom-etry' (EP/N03189X/1)., and ANR-11-IDEX-0001,Amidex,INITIATIVE D'EXCELLENCE AIX MARSEILLE UNIVERSITE(2011)

Subjects
Pure mathematics, Vector bundle, Fano plane, Rational resolutions of singularities, Kempf collapsings, 01 natural sciences, Theoretical Computer Science, Mathematics - Algebraic Geometry, Mathematics (miscellaneous), Morphism, Mathematics::Algebraic Geometry, 0103 physical sciences, FOS: Mathematics, Degeneracy (biology), 0101 mathematics, Algebraic number, Algebraic Geometry (math.AG), Mathematics::Symplectic Geometry, Mathematics, 010102 general mathematics, Degeneracy loci, Calabi-Yau varieties, Orbit closure, Nilpotent, Fano varieties, MSC Primary: 14N05, Secondary: 14E15, 14J32, 14J45, 14M12, Sheaf, 010307 mathematical physics, [MATH.MATH-AG]Mathematics [math]/Algebraic Geometry [math.AG]
Abstract

Degeneracy loci of morphisms between vector bundles have been used in a wide variety of situations. We introduce a vast generalization of this notion, based on orbit closures of algebraic groups in their linear representations. A preferred class of our orbital degeneracy loci is characterized by a certain crepancy condition on the orbit closure, that allows to get some control on the canonical sheaf. This condition is fulfilled for Richardson nilpotent orbits, and also for partially decomposable skew-symmetric three-forms in six variables. In order to illustrate the efficiency and flexibility of our methods, we construct in both situations many Calabi--Yau manifolds of dimension three and four, as well as a few Fano varieties, including some new Fano fourfolds., Comment: To appear in Ann. Sc. Norm. Super. Pisa Cl. Sci. (5)

Published
2019

19. Harnack inequality for kinetic Fokker-Planck equations with rough coefficients and application to the Landau equation

Author

Golse, F, Imbert, Cyril, Mouhot, Clément, Vasseur, A, Centre de Mathématiques Laurent Schwartz (CMLS), École polytechnique (X)-Centre National de la Recherche Scientifique (CNRS), Département de Mathématiques et Applications - ENS Paris (DMA), École normale supérieure - Paris (ENS-PSL), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS), Department of Pure Mathematics and Mathematical Statistics (DPMMS), Faculty of mathematics Centre for Mathematical Sciences [Cambridge] (CMS), University of Cambridge [UK] (CAM)-University of Cambridge [UK] (CAM), Departement of Mathematics [Austin], University of Texas at Austin [Austin], European Project: 279600,EC:FP7:ERC,ERC-2011-StG_20101014,MATKIT(2011), Centre National de la Recherche Scientifique (CNRS)-École polytechnique (X), Centre National de la Recherche Scientifique (CNRS)-École normale supérieure - Paris (ENS Paris), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL), and École normale supérieure - Paris (ENS Paris)

Subjects
Hölder condition, 01 natural sciences, Theoretical Computer Science, Mathematics (miscellaneous), Mathematics - Analysis of PDEs, 35H10, 35B65, ultraparabolic equations, De~Giorgi method, FOS: Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], 0101 mathematics, Landau equation, Mathematics, Harnack's inequality, Hölder continuity, Moser iteration, Operator (physics), 010102 general mathematics, Mathematical analysis, Fokker-Planck equation, Kolmogorov equation, 010101 applied mathematics, Elliptic operator, Hypoelliptic operator, Bounded function, kinetic theory, averaging lemma, Fokker–Planck equation, Linear equation, Analysis of PDEs (math.AP)
Abstract

We extend the De Giorgi--Nash--Moser theory to a class of kinetic Fokker-Planck equations and deduce new results on the Landau-Coulomb equation. More precisely, we first study the H{\"o}lder regularity and establish a Harnack inequality for solutions to a general linear equation of Fokker-Planck type whose coefficients are merely measurable and essentially bounded, i.e. assuming no regularity on the coefficients in order to later derive results for non-linear problems. This general equation has the formal structure of the hypoelliptic equations "of type II" , sometimes also called ultraparabolic equations of Kolmogorov type, but with rough coefficients: it combines a first-order skew-symmetric operator with a second-order elliptic operator involving derivatives along only part of the coordinates and with rough coefficients. These general results are then applied to the non-negative essentially bounded weak solutions of the Landau equation with inverse-power law $\gamma$ $\in$ [--d, 1] whose mass, energy and entropy density are bounded and mass is bounded away from 0, and we deduce the H{\"o}lder regularity of these solutions., Comment: 31 pages, 4 figures

Published
2019

20. A chain rule for the quantum relative entropy

Author

Omar Fawzi, Kun Fang, David Sutter, Renato Renner, Department of Applied Mathematics and Theoretical Physics [Cambridge] (DAMTP), Faculty of mathematics Centre for Mathematical Sciences [Cambridge] (CMS), University of Cambridge [UK] (CAM)-University of Cambridge [UK] (CAM), Laboratoire de l'Informatique du Parallélisme (LIP), École normale supérieure de Lyon (ENS de Lyon)-Université Claude Bernard Lyon 1 (UCBL), Université de Lyon-Université de Lyon-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre National de la Recherche Scientifique (CNRS), Eidgenössische Technische Hochschule - Swiss Federal Institute of Technology [Zürich] (ETH Zürich), ANR-18-CE47-0011,ACOM,Une Théorie Algorithmique de la Communcation(2018), Centre National de la Recherche Scientifique (CNRS)-Université de Lyon-Institut National de Recherche en Informatique et en Automatique (Inria)-Université Claude Bernard Lyon 1 (UCBL), Université de Lyon-École normale supérieure - Lyon (ENS Lyon), and École normale supérieure - Lyon (ENS Lyon)-Université Claude Bernard Lyon 1 (UCBL)

Subjects
FOS: Computer and information sciences, Quantum Physics, Chain rule (probability), Kullback–Leibler divergence, Computer Science - Information Theory, Information Theory (cs.IT), General Physics and Astronomy, TheoryofComputation_GENERAL, FOS: Physical sciences, Context (language use), Conditional probability distribution, Quantum channel, Data_CODINGANDINFORMATIONTHEORY, 01 natural sciences, Quantum relative entropy, Multipartite, [PHYS.QPHY]Physics [physics]/Quantum Physics [quant-ph], 0103 physical sciences, Probability distribution, Statistical physics, 010306 general physics, Quantum Physics (quant-ph), Mathematics
Abstract

The chain rule for the classical relative entropy ensures that the relative entropy between probability distributions on multipartite systems can be decomposed into a sum of relative entropies of suitably chosen conditional distributions on the individual systems. Here, we prove a similar chain rule inequality for the quantum relative entropy in terms of channel relative entropies. The new chain rule allows us to solve an open problem in the context of asymptotic quantum channel discrimination: surprisingly, adaptive protocols cannot improve the error rate for asymmetric channel discrimination compared to non-adaptive strategies. In addition, we give examples of quantum channels showing that the channel relative entropy is not additive under the tensor product., Comment: 12 pages

Published
2019
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21. A master space for moduli spaces of Gieseker-stable sheaves

Author

Matei Toma, Daniel Greb, Julius Ross, Department of Pure Mathematics and Mathematical Statistics (DPMMS), Faculty of mathematics Centre for Mathematical Sciences [Cambridge] (CMS), University of Cambridge [UK] (CAM)-University of Cambridge [UK] (CAM), Institut Élie Cartan de Lorraine (IECL), and Université de Lorraine (UL)-Centre National de la Recherche Scientifique (CNRS)

Subjects
Pure mathematics, Algebra and Number Theory, 010102 general mathematics, Dimension (graph theory), Space (mathematics), Surface (topology), 01 natural sciences, Interpretation (model theory), Moduli space, Mathematics - Algebraic Geometry, Mathematics::Algebraic Geometry, 0103 physical sciences, Mathematik, FOS: Mathematics, 14D20, 14J60, 14L24, 16G20, 010307 mathematical physics, Geometry and Topology, Geometric invariant theory, 0101 mathematics, [MATH]Mathematics [math], Algebraic Geometry (math.AG), Finite set, Quotient, ComputingMilieux_MISCELLANEOUS, Mathematics
Abstract

We consider a notion of stability for sheaves, which we call multi-Gieseker stability that depends on several ample polarisations $L_1, \dots, L_N$ and on an additional parameter $\sigma \in \mathbb{Q}_{\geq 0}^N\setminus\{0\}$. The set of semi stable sheaves admits a projective moduli space $\mathcal M_{\sigma}$. We prove that given a finite collection of parameters $\sigma$, there exists a sheaf- and representation-theoretically defined master space $Y$ such that each corresponding moduli space is obtained from $Y$ as a Geometric Invariant Theory (GIT) quotient. In particular, any two such spaces are related by a finite number of "Thaddeus-flips". As a corollary, we deduce that any two Gieseker-moduli space of sheaves (with respect to different polarisations $L_1$ and $L_2$) are related via a GIT-master space. This confirms an old expectation and generalises results from the surface case to arbitrary dimension., Comment: 18 pages

Published
2019

22. On the Mean Field and Classical Limits of Quantum Mechanics

Author

Thierry Paul, François Golse, Clément Mouhot, Apollo - University of Cambridge Repository, Centre de Mathématiques Laurent Schwartz (CMLS), École polytechnique (X)-Centre National de la Recherche Scientifique (CNRS), Department of Pure Mathematics and Mathematical Statistics (DPMMS), Faculty of mathematics Centre for Mathematical Sciences [Cambridge] (CMS), and University of Cambridge [UK] (CAM)-University of Cambridge [UK] (CAM)

Subjects
Quantum dynamics, [PHYS.MPHY]Physics [physics]/Mathematical Physics [math-ph], math-ph, FOS: Physical sciences, Schrödinger equation, 01 natural sciences, Classical limit, symbols.namesake, Mathematics - Analysis of PDEs, Vlasov equation, math.MP, Hartree equation, Quantum mechanics, 0103 physical sciences, FOS: Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Monge-Kantorovich distance, 0101 mathematics, math.AP, Mathematical Physics, Mean field limit, MSC 82C10, 35Q55 (82C05,35Q83), 82C10, 35Q55, 82C05, 35Q83, Mathematics, 010102 general mathematics, Statistical and Nonlinear Physics, Mathematical Physics (math-ph), BBGKY hierarchy, Second quantization, Liouville equation, Mean field theory, symbols, 010307 mathematical physics, Analysis of PDEs (math.AP), Identical particles
Abstract

The main result in this paper is a new inequality bearing on solutions of the $N$-body linear Schr\"{o}dinger equation and of the mean field Hartree equation. This inequality implies that the mean field limit of the quantum mechanics of $N$ identical particles is uniform in the classical limit and provides a quantitative estimate of the quality of the approximation. This result applies to the case of $C^{1,1}$ interaction potentials. The quantity measuring the approximation of the $N$-body quantum dynamics by its mean field limit is analogous to the Monge-Kantorovich (or Wasserstein) distance with exponent $2$. The inequality satisfied by this quantity is reminiscent of the work of Dobrushin on the mean field limit in classical mechanics [Func. Anal. Appl. 13 (1979), 115-123]. Our approach of this problem is based on a direct analysis of the $N$-particle Liouville equation, and avoids using techniques based on the BBGKY hierarchy or on second quantization., Comment: 38 pages. Final version, to appear in Communications in Mathematical Physics

Published
2016

23. Cacti and filtered distributive laws

Author

Vladimir Dotsenko, James M. Griffin, Trinity College Dublin, Department of Pure Mathematics and Mathematical Statistics (DPMMS), Faculty of mathematics Centre for Mathematical Sciences [Cambridge] (CMS), and University of Cambridge [UK] (CAM)-University of Cambridge [UK] (CAM)

Subjects
Classifying space, Pure mathematics, Coalgebra, Homology (mathematics), Mathematics::Algebraic Topology, Ground field, distributive law, Koszul operad, 18D50, Mathematics::K-Theory and Homology, Mathematics::Quantum Algebra, Mathematics::Category Theory, FOS: Mathematics, Algebraic Topology (math.AT), [INFO]Computer Science [cs], Mathematics - Algebraic Topology, [MATH]Mathematics [math], based cactus products, Mathematics, K-Theory and Homology (math.KT), 16. Peace & justice, Automorphism, 20L05, 16S15, Distributive property, Free product, Mathematics - K-Theory and Homology, Simplicial set, Geometry and Topology, Gröbner basis, 55P48 (Primary), 16S15, 18D50, 20L05 (Secondary)
Abstract

Motivated by the second author's construction of a classifying space for the group of pure symmetric automorphisms of a free product, we introduce and study a family of topological operads, the operads of based cacti, defined for every pointed topological space $(Y,\bullet)$. These operads also admit linear versions, which are defined for every augmented graded cocommutative coalgebra $C$. We show that the homology of the topological operad of based $Y$-cacti is the linear operad of based $H_*(Y)$-cacti. In addition, we show that for every coalgebra $C$ the operad of based $C$-cacti is Koszul. To prove the latter result, we use the criterion of Koszulness for operads due to the first author, utilising the notion of a filtered distributive law between two quadratic operads. We also present a new proof of that criterion which works over the ground field of arbitrary characteristic., 30 pages

Published
2014

24. The quasineutral limit of the Vlasov–Poisson equation in Wasserstein metric

Author

Mikaela Iacobelli, Daniel Han-Kwan, Centre de Mathématiques Laurent Schwartz (CMLS), École polytechnique (X)-Centre National de la Recherche Scientifique (CNRS), Department of Pure Mathematics and Mathematical Statistics (DPMMS), Faculty of mathematics Centre for Mathematical Sciences [Cambridge] (CMS), and University of Cambridge [UK] (CAM)-University of Cambridge [UK] (CAM)

Subjects
Work (thermodynamics), Applied Mathematics, General Mathematics, 010102 general mathematics, Mathematical analysis, 01 natural sciences, Measure (mathematics), Stability (probability), 010101 applied mathematics, Massless particle, Wasserstein metric, Convergence (routing), [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Limit (mathematics), 0101 mathematics, Poisson's equation, ComputingMilieux_MISCELLANEOUS, Mathematics
Abstract

In this work, we study the quasineutral limit of the one-dimensional Vlasov-Poisson equation for ions with massless thermalized electrons. We prove new weak-strong stability estimates in the Wasserstein metric that allow us to extend and improve previously known convergence results. In particular, we show that given a possibly unstable analytic initial profile, the formal limit holds for sequences of measure initial data converging sufficiently fast in the Wasserstein metric to this profile. This is achieved without assuming uniform analytic regularity.

Published
2017

25. Cell motility in confinement: a computational model for the shape of the cell

Author

Julien Olivier, Meriem Jedouaa, Olivier Theodoly, Florence Hubert, Ariane Trescases, I. Khames, Institut de Mathématiques de Marseille (I2M), Aix Marseille Université (AMU)-École Centrale de Marseille (ECM)-Centre National de la Recherche Scientifique (CNRS), Equations aux Dérivées Partielles (EDP ), Laboratoire Jean Kuntzmann (LJK ), Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP )-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes [2016-2019] (UGA [2016-2019])-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP )-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes [2016-2019] (UGA [2016-2019]), Laboratoire de Mathématiques de l'INSA de Rouen Normandie (LMI), Institut national des sciences appliquées Rouen Normandie (INSA Rouen Normandie), Institut National des Sciences Appliquées (INSA)-Normandie Université (NU)-Institut National des Sciences Appliquées (INSA)-Normandie Université (NU), Adhésion et Inflammation (LAI), Aix Marseille Université (AMU)-Assistance Publique - Hôpitaux de Marseille (APHM)-Institut National de la Santé et de la Recherche Médicale (INSERM)-Centre National de la Recherche Scientifique (CNRS), Department of Pure Mathematics and Mathematical Statistics (DPMMS), Faculty of mathematics Centre for Mathematical Sciences [Cambridge] (CMS), University of Cambridge [UK] (CAM)-University of Cambridge [UK] (CAM), Assistance Publique - Hôpitaux de Marseille (APHM)-Institut National de la Santé et de la Recherche Médicale (INSERM)-Aix Marseille Université (AMU)-Centre National de la Recherche Scientifique (CNRS), HUBERT, Florence, and Centre National de la Recherche Scientifique (CNRS)-École Centrale de Marseille (ECM)-Aix Marseille Université (AMU)

Subjects
Physics, T57-57.97, Applied mathematics. Quantitative methods, Microchannel, Cell, Motility, Nanotechnology, macromolecular substances, Mechanics, medicine.anatomical_structure, Cytoplasm, QA1-939, medicine, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], [MATH.MATH-AP] Mathematics [math]/Analysis of PDEs [math.AP], Lamellipodium, Mathematics
Abstract

While cells typically tend to spread their cytoplasm in a flat and thin lamellipodium when moving on a flat substrate, it is widely observed that the cytoplasm has a compact shape in micro-channels, tending to fulfill the cross-section of the microchannel. We propose a minimal mathematical model for a 2D test case which describes the cell lamellipodium deformations when confined in a channel. We then go through a numerical investigation of this mathematical model and show that it allows to recover qualitatively the physiological characteristics of the confined cell., Alors que les cellules ont généralement tendance à présenter un cytoplasme étendu en un large lamellipode extrêmement fin lors du déplacement sur un substrat plat, il est communément observé que le cytoplasme prend une forme compacte lors du déplacement dans des micro-canaux, remplissant au possible le volume contenu dans le micro-channel. Nous proposons un modèle mathématique minimal pour un cas test en 2D qui décrit les déformations du lamellipode en confinement. Nous proposons une exploration numérique de ce modèle mathématique et nous montrons qu’il permet de retrouver qualitativement les caractéristiques physiologiques de la cellule confinée.

Published
2016

26. Empirical measures and Vlasov hierarchies

Author

Clément Mouhot, Valeria Ricci, François Golse, Golse, F, Mouhot, C, Ricci, V, Centre de Mathématiques Laurent Schwartz (CMLS), École polytechnique (X)-Centre National de la Recherche Scientifique (CNRS), Department of Pure Mathematics and Mathematical Statistics (DPMMS), Faculty of mathematics Centre for Mathematical Sciences [Cambridge] (CMS), University of Cambridge [UK] (CAM)-University of Cambridge [UK] (CAM), Dipartimento di Matematica e Informatica, and Università degli studi di Palermo - University of Palermo

Subjects
MSC 82C05 (35F25, 28A33), [PHYS.MPHY]Physics [physics]/Mathematical Physics [math-ph], FOS: Physical sciences, 01 natural sciences, Vlasov type equation, Mean-field limit, Empirical measure, BBGKY hierarchy, Monge-Kantorovich distance, 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], Applied mathematics, Monge-Kantorovich distance, Direct proof, 0101 mathematics, Empirical measure, Mathematical Physics, Mean field limit, Mathematics, Numerical Analysis, Hierarchy, 010102 general mathematics, Vlasov type equation, Mathematical Physics (math-ph), BBGKY hierarchy, Lipschitz continuity, 010101 applied mathematics, Kernel (algebra), Uniqueness theorem for Poisson's equation, Modeling and Simulation, Exponent, 82C05 (35F25, 28A33), Analysis of PDEs (math.AP)
Abstract

The present note reviews some aspects of the mean field limit for Vlasov type equations with Lipschitz continuous interaction kernel. We discuss in particular the connection between the approach involving the N-particle empirical measure and the formulation based on the BBGKY hierarchy. This leads to a more direct proof of the quantitative estimates on the propagation of chaos obtained on a more general class of interacting systems in [S.Mischler, C. Mouhot, B. Wennberg, arXiv:1101.4727]. Our main result is a stability estimate on the BBGKY hierarchy uniform in the number of particles, which implies a stability estimate in the sense of the Monge-Kantorovich distance with exponent 1 on the infinite mean field hierarchy. This last result amplifies Spohn's uniqueness theorem [H. Spohn, Math. Meth. Appl. Sci. 3 (1981), 445-455]., 25 pages, one typo corrected, some explanations on the notion of statistical solution of the mean field PDE added in sections 5-6, some comments on the meaning of Theorem 3.1 added at the beginning of section 3, one reference added

Published
2013

27. Geometric properties of solutions to the total variation denoising problem

Author

Vincent Duval, Gabriel Peyré, Clarice Poon, Antonin Chambolle, Centre de Mathématiques Appliquées - Ecole Polytechnique (CMAP), École polytechnique (X)-Centre National de la Recherche Scientifique (CNRS), CEntre de REcherches en MAthématiques de la DEcision (CEREMADE), Université Paris Dauphine-PSL, Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS), Méthodes numériques pour le problème de Monge-Kantorovich et Applications en sciences sociales (MOKAPLAN), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Dauphine-PSL, Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS)-Inria de Paris, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Département de Mathématiques et Applications - ENS Paris (DMA), École normale supérieure - Paris (ENS-PSL), Department of Applied Mathematics and Theoretical Physics [Cambridge] (DAMTP), Faculty of mathematics Centre for Mathematical Sciences [Cambridge] (CMS), University of Cambridge [UK] (CAM)-University of Cambridge [UK] (CAM), European Project: 279593,EC:FP7:ERC,ERC-2011-StG_20101014,SIGMA-VISION(2011), Centre National de la Recherche Scientifique (CNRS)-Université Paris Dauphine-PSL, Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Inria de Paris, Centre National de la Recherche Scientifique (CNRS)-École normale supérieure - Paris (ENS Paris), Inria de Paris, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-CEntre de REcherches en MAthématiques de la DEcision (CEREMADE), and École normale supérieure - Paris (ENS Paris)

Subjects
Noise reduction, 02 engineering and technology, Classification of discontinuities, 01 natural sciences, Regularization (mathematics), Upper and lower bounds, Theoretical Computer Science, [INFO.INFO-TS]Computer Science [cs]/Signal and Image Processing, FOS: Mathematics, 0202 electrical engineering, electronic engineering, information engineering, denoising, 0101 mathematics, Mathematics - Optimization and Control, Mathematical Physics, Mathematics, calibrable set, Total variation, Applied Mathematics, Image regularization, 010102 general mathematics, Regular polygon, Total variation denoising, Computer Science Applications, Rate of convergence, Optimization and Control (math.OC), Signal Processing, 020201 artificial intelligence & image processing, [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], Algorithm, optimization
Abstract

International audience; This article studies the denoising performance of total variation (TV) image regularization. More precisely, we study geometrical properties of the solution to the so-called Rudin-Osher-Fatemi total variation denoising method. The first contribution of this paper is a precise mathematical definition of the “extended support” (associated to the noise-free image) of TV denoising. It is intuitively the region which is unstable and will suffer from the staircasing effect. We highlight in several practical cases, such as the indicator of convex sets, that this region can be determined explicitly. Our second and main contribution is a proof that the TV denoising method indeed restores an image which is exactly constant outside a small tube surrounding the extended support. The radius of this tube shrinks toward zero as the noise level vanishes, and are able to determine, in some cases, an upper bound on the convergence rate. For indicators of so-called “calibrable” sets (such as disks or properly eroded squares), this extended support matches the edges, so that discontinuities produced by TV denoising cluster tightly around the edges. In contrast, for indicators of more general shapes or for complicated images, this extended support can be larger. Beside these main results, our paper also proves several intermediate results about fine properties of TV regularization, in particular for indicators of calibrable and convex sets, which are of independent interest.

Published
2016

28. A spectral characterization of nonlinear normal modes

Author

Gaëtan Kerschen, Rodolphe Sepulchre, Giuseppe Ilario Cirillo, Ludovic Renson, Alexandre Mauroy, Université de Liège, Department of Engineering Mathematics, University of Bristol [Bristol], Space Structures and Systems Lab, Department of Aerospace and Mechanical Engineering, Université de Liège-Université de Liège, Control Group, Department of Engineering, University of Cambridge [UK] (CAM)-University of Cambridge [UK] (CAM), Cirillo, GI [0000-0002-1162-3225], and Apollo - University of Cambridge Repository

Subjects
invariant manifolds, Acoustics and Ultrasonics, Invariant manifold, Dynamical Systems (math.DS), 01 natural sciences, 010305 fluids & plasmas, [SPI]Engineering Sciences [physics], Normal mode, 0103 physical sciences, FOS: Mathematics, characterization, Mathematics - Dynamical Systems, Invariant (mathematics), [MATH]Mathematics [math], Nonlinear normal modes, Koopman operator, 010301 acoustics, Mathematics, Spectral, Parametrization, Mechanical Engineering, Operator (physics), Mathematical analysis, nonlinear normal modes, parametrization, Eigenfunction, Condensed Matter Physics, Nonlinear system, Invariant manifolds, Mechanics of Materials, Phase space, spectral
Abstract

This paper explores the relationship that exists between nonlinear normal modes (NNMs) defined as invariant manifolds in phase space and the spectral expansion of the Koopman operator. Specifically, we demonstrate that NNMs correspond to zero level sets of specific eigenfunctions of the Koopman operator. Thanks to this direct connection, a new, global parametrization of the invariant manifolds is established. Unlike the classical parametrization using a pair of state-space variables, this parametrization remains valid whenever the invariant manifold undergoes folding, which extends the computation of NNMs to regimes of greater energy. The proposed ideas are illustrated using a two-degree-of-freedom system with cubic nonlinearity., Belgian Network DYSCO (Dynamical Systems, Control, and Optimization) funded by the Interuniversity Attraction Poles Programme initiated by the Belgian Science Policy Office, This is the author accepted manuscript. The final version is available from Elsevier via http://dx.doi.org/10.1016/j.jsv.2016.05.016

Published
2016
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29. Total Variation Denoising and Support Localization of the Gradient

Author

Antonin Chambolle, Gabriel Peyré, Vincent Duval, Clarice Poon, Centre de Mathématiques Appliquées - Ecole Polytechnique (CMAP), École polytechnique (X)-Centre National de la Recherche Scientifique (CNRS), Méthodes numériques pour le problème de Monge-Kantorovich et Applications en sciences sociales (MOKAPLAN), CEntre de REcherches en MAthématiques de la DEcision (CEREMADE), Université Paris Dauphine-PSL, Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Dauphine-PSL, Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS)-Inria de Paris, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS), Department of Applied Mathematics and Theoretical Physics [Cambridge] (DAMTP), Faculty of mathematics Centre for Mathematical Sciences [Cambridge] (CMS), University of Cambridge [UK] (CAM)-University of Cambridge [UK] (CAM), and IOP

Subjects
History, Mathematical optimization, Noise reduction, Radius, Total variation denoising, 01 natural sciences, Upper and lower bounds, Computer Science Applications, Education, Image (mathematics), Zero (linguistics), 010101 applied mathematics, Rate of convergence, 0101 mathematics, [MATH]Mathematics [math], Constant (mathematics), Algorithm, Mathematics
Abstract

This paper describes the geometrical properties of the solutions to the total variation denoising method. A folklore statement is that this method is able to restore sharp edges, but at the same time, might introduce some staircasing (i.e. "fake" edges) in flat areas. Quite surprisingly, put aside numerical evidences, almost no theoretical result are available to backup these claims. The first contribution of this paper is a precise mathematical definition of the "extended support" (associated to the noise-free image) of TV denoising. This is intuitively the region which is unstable and will suffer from the staircasing effect. Our main result shows that the TV denoising method indeed restores a piece-wise constant image outside a small tube surrounding the extended support. Furthermore, the radius of this tube shrinks toward zero as the noise level vanishes and in some cases, an upper bound on the convergence rate is given.

Published
2016

30. Capacity of the range of random walk on $\mathbb Z^4$

Author

Bruno Schapira, Perla Sousi, Amine Asselah, Schapira, Bruno, Marches aléatoires en interaction - - MALIN2016 - ANR-16-CE93-0003 - AAPG2016 - VALID, Laboratoire d'Analyse et de Mathématiques Appliquées (LAMA), Université Paris-Est Marne-la-Vallée (UPEM)-Fédération de Recherche Bézout-Université Paris-Est Créteil Val-de-Marne - Paris 12 (UPEC UP12)-Centre National de la Recherche Scientifique (CNRS), Institut de Mathématiques de Marseille (I2M), Centre National de la Recherche Scientifique (CNRS)-École Centrale de Marseille (ECM)-Aix Marseille Université (AMU), DPMMS/CMS, University of Cambridge [UK] (CAM), ANR-16-CE93-0003,MALIN,Marches aléatoires en interaction(2017), Centre National de la Recherche Scientifique (CNRS)-Université Paris-Est Créteil Val-de-Marne - Paris 12 (UPEC UP12)-Fédération de Recherche Bézout-Université Paris-Est Marne-la-Vallée (UPEM), Aix Marseille Université (AMU)-École Centrale de Marseille (ECM)-Centre National de la Recherche Scientifique (CNRS), Department of Pure Mathematics and Mathematical Statistics (DPMMS), Faculty of mathematics Centre for Mathematical Sciences [Cambridge] (CMS), University of Cambridge [UK] (CAM)-University of Cambridge [UK] (CAM), and ANR-16-CE93-0003,MALIN,Marches aléatoires en interaction(2016)

Subjects
Statistics and Probability, 60G50, [MATH.MATH-PR] Mathematics [math]/Probability [math.PR], Central limit theorem, Integer lattice, Law of large numbers, 01 natural sciences, 010104 statistics & probability, Mathematics::Probability, Dimension (vector space), 60F05, FOS: Mathematics, 60F50, Limit (mathematics), 0101 mathematics, Mathematics, Discrete mathematics, Capacity, Probability (math.PR), 010102 general mathematics, Random walk, Green kernel, [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], Range (mathematics), Scaling limit, Statistics, Probability and Uncertainty, Mathematics - Probability
Abstract

We study the scaling limit of the capacity of the range of a random walk on the integer lattice in dimension four. We establish a strong law of large numbers and a central limit theorem with a non-Gaussian limit. The asymptotic behaviour is analogous to that found by Le Gall in ’86 [Comm. Math. Phys. 104 (1986) 471–507] for the volume of the range in dimension two.

Published
2016

31. Estimation of Rotor Position and Speed of Permanent Magnet Synchronous Motors With Guaranteed Stability

Author

Kwanghee Nam, Romeo Ortega, Junggi Lee, Laurent Praly, Alessandro Astolfi, Laboratoire des signaux et systèmes (L2S), Université Paris-Sud - Paris 11 (UP11)-CentraleSupélec-Centre National de la Recherche Scientifique (CNRS), Centre Automatique et Systèmes (CAS), MINES ParisTech - École nationale supérieure des mines de Paris, Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL), EEE Department, Imperial College London, NCAS-Climate [Cambridge], Department of Chemistry [Cambridge, UK], University of Cambridge [UK] (CAM)-University of Cambridge [UK] (CAM), Department of Electrical Engineering, and Pohang University of Science and Technology (POSTECH)

Subjects
0209 industrial biotechnology, Observer (quantum physics), Automatic control, observer design, 02 engineering and technology, Nonlinear control, [SPI.AUTO]Engineering Sciences [physics]/Automatic, law.invention, 020901 industrial engineering & automation, Settore ING-INF/04 - Automatica, Motor control, Control theory, Position (vector), law, [INFO.INFO-AU]Computer Science [cs]/Automatic Control Engineering, 0202 electrical engineering, electronic engineering, information engineering, Electrical and Electronic Engineering, Machine control, Mathematics, Rotor (electric), 020208 electrical & electronic engineering, stability, Motor control, nonlinear control, observer design, stability, Rate of convergence, Control and Systems Engineering, nonlinear control, Synchronous motor
Abstract

International audience; The control algorithms used in high performance ac drives require the knowledge of rotor position and, in the case of speed regulation, also of speed. Since in many applications rotational transducers cannot be installed, their reconstruction is needed. The use of observers is stymied by the fact that the dynamics of electrical machines are highly nonlinear and does not belong to the class studied by the nonlinear control community. In this paper solutions to both problems, which are particularly tailored for the widely popular permanent magnet synchronous motors, are provided. A key step for the design of both observers is the choice of a suitable set of coordinates. The position observer is a standard gradient search whose detailed analysis reveals outstanding (global asymptotic) stability properties. Furthermore, the analysis clearly exhibits the interplay between rotor speed and the gain of the gradient search--that (essentially) determines its convergence rate. The position observer is a simple two-dimensional nonlinear system, hence is easily implementable. The speed observer is designed following the immersion and invariance technique and is also shown to be globally convergent. Simulation and experimental results of the position observer, used together with a classical field-oriented control algorithm, are presented

Published
2011

32. Linear Forms and Higher-Degree Uniformity for Functions On $${\mathbb{F}^{n}_{p}}$$

Author

W.T. Gowers, J. Wolf, Juppin, Carole, Department of Pure Mathematics and Mathematical Statistics (DPMMS), Faculty of mathematics Centre for Mathematical Sciences [Cambridge] (CMS), University of Cambridge [UK] (CAM)-University of Cambridge [UK] (CAM), Department of Mathematics - Rutgers School of Arts and Sciences, Rutgers, The State University of New Jersey [New Brunswick] (RU), and Rutgers University System (Rutgers)-Rutgers University System (Rutgers)

Subjects
Polynomial, Inverse, 0102 computer and information sciences, 01 natural sciences, Combinatorics, [MATH.MATH-CO]Mathematics [math]/Combinatorics [math.CO], FOS: Mathematics, Mathematics - Combinatorics, Number Theory (math.NT), 0101 mathematics, Mathematics, 11B30, Conjecture, Mathematics - Number Theory, Degree (graph theory), 010102 general mathematics, Order (ring theory), [MATH.MATH-NT]Mathematics [math]/Number Theory [math.NT], [MATH.MATH-CO] Mathematics [math]/Combinatorics [math.CO], 010201 computation theory & mathematics, Bounded function, Norm (mathematics), Combinatorics (math.CO), Geometry and Topology, Linear independence, Analysis, [MATH.MATH-NT] Mathematics [math]/Number Theory [math.NT]
Abstract

In [GW09a] we conjectured that uniformity of degree $k-1$ is sufficient to control an average over a family of linear forms if and only if the $k$th powers of these linear forms are linearly independent. In this paper we prove this conjecture in $\mathbb{F}_p^n$, provided only that $p$ is sufficiently large. This result represents one of the first applications of the recent inverse theorem for the $U^k$ norm over $\mathbb{F}_p^n$ by Bergelson, Tao and Ziegler [BTZ09,TZ08]. We combine this result with some abstract arguments in order to prove that a bounded function can be expressed as a sum of polynomial phases and a part that is small in the appropriate uniformity norm. The precise form of this decomposition theorem is critical to our proof, and the theorem itself may be of independent interest., 40 pages

Published
2011

33. Critical branching Brownian motion with absorption: Particle configurations

Author

Julien Berestycki, Jason Schweinsberg, Nathanaël Berestycki, 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), Department of Pure Mathematics and Mathematical Statistics (DPMMS), Faculty of mathematics Centre for Mathematical Sciences [Cambridge] (CMS), University of Cambridge [UK] (CAM)-University of Cambridge [UK] (CAM), Department of Mathematics [Univ California San Diego] (MATH - UC San Diego), University of California [San Diego] (UC San Diego), University of California (UC)-University of California (UC), Department of Mathematics at the University of California at San Diego, and University of California-University of California

Subjects
Statistics and Probability, Particle number, Statistics & Probability, 60J65, 60J80, 60J25, Critical phenomena, Branching (polymer chemistry), math.PR, 01 natural sciences, 010104 statistics & probability, 60J25, FOS: Mathematics, 0101 mathematics, ComputingMilieux_MISCELLANEOUS, Brownian motion, Mathematics, 60J80, Statistics, Probability (math.PR), 010102 general mathematics, Mathematical analysis, Primary 60J65, critical phenomena, Yaglom limit laws, Brownian excursion, [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], Extinction time, Branching Brownian motion, Primary 60J65, Secondary 60J80, 60J25, Statistics, Probability and Uncertainty, Mathematics - Probability, Secondary 60J80
Abstract

International audience; We consider critical branching Brownian motion with absorption, in which there is initially a single particle at x > 0, particles move according to independent one-dimensional Brownian motions with the critical drift of -root 2, and particles are absorbed when they reach zero. Here we obtain asymptotic results concerning the behavior of the process before the extinction time, as the position x of the initial particle tends to infinity. We estimate the number of particles in the system at a given time and the position of the right-most particle. We also obtain asymptotic results for the configuration of particles at a typical time.

Published
2015

34. On measure solutions of the Boltzmann equation, Part II: Rate of convergence to equilibrium

Author

Clément Mouhot, Xuguang Lu, Department of Mathematical Sciences, Tsinghua University [Beijing] (THU), Department of Pure Mathematics and Mathematical Statistics (DPMMS), Faculty of mathematics Centre for Mathematical Sciences [Cambridge] (CMS), University of Cambridge [UK] (CAM)-University of Cambridge [UK] (CAM), and European Project: 279600,EC:FP7:ERC,ERC-2011-StG_20101014,MATKIT(2011)

Subjects
global-in-time stability, 01 natural sciences, Measure (mathematics), equilibrium, 010305 fluids & plasmas, Momentum, Boltzmann equation, Mathematics - Analysis of PDEs, 0103 physical sciences, Convergence (routing), Applied mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], measure solutions, exponential rate of convergence, 0101 mathematics, [MATH]Mathematics [math], eternal solution, Mathematics, spatially homogeneous, 35Q [See also 35J05, 35J10, 35K05, 35L05], 76P05 [See also 82B40, 82C40, 82D05], Applied Mathematics, hard potentials, 010102 general mathematics, Conserved quantity, Exponential function, Rate of convergence, Iterated function, Analysis
Abstract

The paper considers the convergence to equilibrium for measure solutions of the spatially homogeneous Boltzmann equation for hard potentials with angular cutoff. We prove the exponential sharp rate of strong convergence to equilibrium for conservative measure solutions having finite mass and energy. The proof is based on the regularizing property of the iterated collision operators, exponential moment production estimates, and some previous results on the exponential rate of strong convergence to equilibrium for square integrable initial data. We also obtain a lower bound of the convergence rate and deduce that no eternal solutions exist apart from the trivial stationary solutions given by the Maxwellian equilibrium. We finally use these convergence rates in order to deduce global-in-time strong stability of measure solutions., Comment: 60 pages -- some typos corrected and explanations added in the new version

Published
2015

35. Partitions and orientations of the Rado graph

Author

Stéphan Thomassé, Reinhard Diestel, Imre Leader, Alex Scott, Department Mathematik, Universität Hamburg (UHH), Department of Pure Mathematics and Mathematical Statistics (DPMMS), Faculty of mathematics Centre for Mathematical Sciences [Cambridge] (CMS), University of Cambridge [UK] (CAM)-University of Cambridge [UK] (CAM), Oxford Combinatorics, University of Oxford [Oxford], Algorithmes, Graphes et Combinatoire (ALGCO), Laboratoire d'Informatique de Robotique et de Microélectronique de Montpellier (LIRMM), and Centre National de la Recherche Scientifique (CNRS)-Université de Montpellier (UM)-Centre National de la Recherche Scientifique (CNRS)-Université de Montpellier (UM)

Subjects
Discrete mathematics, Applied Mathematics, General Mathematics, Symmetric graph, 010102 general mathematics, Voltage graph, Comparability graph, 0102 computer and information sciences, 01 natural sciences, law.invention, Combinatorics, Rado graph, Computer Science::Discrete Mathematics, 010201 computation theory & mathematics, law, [MATH.MATH-CO]Mathematics [math]/Combinatorics [math.CO], Line graph, Cubic graph, 0101 mathematics, Null graph, Universal graph, Mathematics
Abstract

International audience; We classify the countably infinite oriented graphs which, for every partition of their vertex set into two parts, induce an isomorphic copy of themselves on at least one of the parts. These graphs are the edgeless graph, the random tournament, the transitive tournaments of order type~$\omega^\alpha$, and two orientations of the Rado graph: the random oriented graph, and a newly found random acyclic oriented graph.

Published
2006

36. Critical branching Brownian motion with absorption: survival probability

Author

Julien Berestycki, Nathanaël Berestycki, Jason Schweinsberg, 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), Department of Pure Mathematics and Mathematical Statistics (DPMMS), Faculty of mathematics Centre for Mathematical Sciences [Cambridge] (CMS), University of Cambridge [UK] (CAM)-University of Cambridge [UK] (CAM), Department of Mathematics at the University of California at San Diego, University of California [San Diego] (UC San Diego), University of California-University of California, Department of Mathematics [Univ California San Diego] (MATH - UC San Diego), and University of California (UC)-University of California (UC)

Subjects
Statistics and Probability, Extinction time, 60J65, 60J80, 60J25, Statistics & Probability, 01 natural sciences, Upper and lower bounds, math.PR, Critical phenomena, 010104 statistics & probability, Mathematics::Probability, 60J25, FOS: Mathematics, Calculus, Almost surely, Statistical physics, 0101 mathematics, Brownian motion, ComputingMilieux_MISCELLANEOUS, Mathematics, Geometric Brownian motion, Fractional Brownian motion, Applied Mathematics, Probability (math.PR), 010102 general mathematics, Statistics, Primary 60J65, Brownian excursion, Pure Mathematics, [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], Branching processes, Reflected Brownian motion, Diffusion process, Branching Brownian motion, Primary 60J65, Secondary 60J80, 60J25, Survival probability, Statistics, Probability and Uncertainty, Mathematics - Probability, Analysis, Secondary 60J80
Abstract

We consider branching Brownian motion on the real line with absorption at zero, in which particles move according to independent Brownian motions with the critical drift of $$-\sqrt{2}$$ . Kesten (Stoch Process 7:9–47, 1978) showed that almost surely this process eventually dies out. Here we obtain upper and lower bounds on the probability that the process survives until some large time $$t$$ . These bounds improve upon results of Kesten (Stoch Process 7:9–47, 1978), and partially confirm nonrigorous predictions of Derrida and Simon (EPL 78:60006, 2007).

Published
2014

37. Extension from Precoloured Sets of Edges

Author

Gregory J. Puleo, Katherine Edwards, Jean-Sébastien Sereni, António Girão, Ross J. Kang, Jan van den Heuvel, Department of Computer Science, Princeton University, Department of Pure Mathematics and Mathematical Statistics (DPMMS), Faculty of mathematics Centre for Mathematical Sciences [Cambridge] (CMS), University of Cambridge [UK] (CAM)-University of Cambridge [UK] (CAM), Department Mathematics [London] (LSE), London School of Economics and Political Science (LSE), Applied Stochastics (IMAPP), Radboud University [Nijmegen], Coordinated Science Laboratory (CSL), University of Illinois System, Knowledge representation, reasonning (ORPAILLEUR), Inria Nancy - Grand Est, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Department of Natural Language Processing & Knowledge Discovery (LORIA - NLPKD), Laboratoire Lorrain de Recherche en Informatique et ses Applications (LORIA), Institut National de Recherche en Informatique et en Automatique (Inria)-Université de Lorraine (UL)-Centre National de la Recherche Scientifique (CNRS)-Institut National de Recherche en Informatique et en Automatique (Inria)-Université de Lorraine (UL)-Centre National de la Recherche Scientifique (CNRS)-Laboratoire Lorrain de Recherche en Informatique et ses Applications (LORIA), Institut National de Recherche en Informatique et en Automatique (Inria)-Université de Lorraine (UL)-Centre National de la Recherche Scientifique (CNRS)-Université de Lorraine (UL)-Centre National de la Recherche Scientifique (CNRS), Ross J. Kang is supported by Veni (639.031.138) and Vidi (639.032.614) grants of the Netherlands Organisation for Scientific Research (NWO).Jean-Sébastien Sereni’ s work was partially supported by Agence Nationale de la Recherche under references ANR-10 -JCJC-0204-01 and ANR-13-BS02-0007., ANR-10-JCJC-0204,HEREDIA,Classes héréditaires de graphes(2010), ANR-13-BS02-0007,Stint,Structures Interdites(2013), and Radboud university [Nijmegen]

Subjects
0102 computer and information sciences, 01 natural sciences, Theoretical Computer Science, mutligraph, Combinatorics, Computer Science::Discrete Mathematics, [MATH.MATH-CO]Mathematics [math]/Combinatorics [math.CO], FOS: Mathematics, Discrete Mathematics and Combinatorics, Mathematics - Combinatorics, QA Mathematics, 0101 mathematics, Mathematics, Conjecture, Applied Mathematics, 010102 general mathematics, edge-colouring, 16. Peace & justice, Graph, Edge coloring, 05C15, Computational Theory and Mathematics, 010201 computation theory & mathematics, colouring extension, precolouring, Geometry and Topology, Combinatorics (math.CO), MathematicsofComputing_DISCRETEMATHEMATICS
Abstract

We consider precolouring extension problems for proper edge-colourings of graphs and multigraphs, in an attempt to prove stronger versions of Vizing's and Shannon's bounds on the chromatic index of (multi)graphs in terms of their maximum degree $\Delta$. We are especially interested in the following question: when is it possible to extend a precoloured matching to a colouring of all edges of a (multi)graph? This question turns out to be related to the notorious List Colouring Conjecture and other classic notions of choosability., Comment: 26 pages, 3 figures, 1 table; in v2, two co-authors added, main conjecture weakened, weak version of conjecture proved, planar section updated; v3 accepted to Electronic Journal of Combinatorics

Published
2014

38. A Note on Chow Stability of the Projectivization of Gieseker Stable Bundles

Author

Julien Keller, Julius Ross, Institut de Mathématiques de Marseille (I2M), Aix Marseille Université (AMU)-École Centrale de Marseille (ECM)-Centre National de la Recherche Scientifique (CNRS), Department of Pure Mathematics and Mathematical Statistics (DPMMS), Faculty of mathematics Centre for Mathematical Sciences [Cambridge] (CMS), University of Cambridge [UK] (CAM)-University of Cambridge [UK] (CAM), and ANR-10-BLAN-0118,MNGNK,Méthodes nouvelles en géométrie non-kählerienne(2010)

Subjects
Mathematics - Differential Geometry, Projectivization, Pure mathematics, Algebraic geometry Geometric invariant theory, Gieseker stability, 14L24, 32Q20, Algebraic geometry, Stability (probability), Mathematics - Algebraic Geometry, Mathematics::Algebraic Geometry, Mathematics::K-Theory and Homology, FOS: Mathematics, Complex Variables (math.CV), Algebraic Geometry (math.AG), Mathematics::Symplectic Geometry, Mathematics, 14L24, 53C25, 32Q20, 53C07, 32Q15, Mathematics - Complex Variables, projectivisation, Mathematical analysis, K-stability, [MATH.MATH-CV]Mathematics [math]/Complex Variables [math.CV], Manifold, Chow stabilty, Differential Geometry (math.DG), Differential geometry, [MATH.MATH-DG]Mathematics [math]/Differential Geometry [math.DG], Bundle, Geometry and Topology, Geometric invariant theory, ruled manifold, [MATH.MATH-AG]Mathematics [math]/Algebraic Geometry [math.AG], Constant scalar curvature Kähler metrics, Scalar curvature
Abstract

We investigate Chow stability of projective bundles P(E) where E is a strictly Gieseker stable bundle over a base manifold that has constant scalar curvature. We show that, for suitable polarisations L, the pair (P(E),L) is Chow stable and give examples for which it is not asymptotically Chow stable., 23 pages. Theorem 2 changed to fix an error in conventions, and the Example in Section 5 rewritten to accommodate. Other minor changes, mostly expositional

Published
2014

39. Martingale defocusing and transience of a self-interacting random walk

Author

Perla Sousi, Yuval Peres, Bruno Schapira, Apollo - University of Cambridge Repository, Theory Group - Microsoft Research, Microsoft Research, Institut de Mathématiques de Marseille (I2M), Aix Marseille Université (AMU)-École Centrale de Marseille (ECM)-Centre National de la Recherche Scientifique (CNRS), Department of Pure Mathematics and Mathematical Statistics (DPMMS), Faculty of mathematics Centre for Mathematical Sciences [Cambridge] (CMS), and University of Cambridge [UK] (CAM)-University of Cambridge [UK] (CAM)

Subjects
Statistics and Probability, Heterogeneous random walk in one dimension, Self-interacting random walk, 010102 general mathematics, Loop-erased random walk, 4901 Applied Mathematics, Probability (math.PR), Random walk, 01 natural sciences, Excited random walk, [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], Combinatorics, 010104 statistics & probability, 4905 Statistics, Martingale, Mathematics::Probability, 60K35, 49 Mathematical Sciences, Transience, FOS: Mathematics, 0101 mathematics, Statistics, Probability and Uncertainty, Martingale (probability theory), Mathematics - Probability, Mathematics
Abstract

Suppose that $(X,Y,Z)$ is a random walk in $\Z^3$ that moves in the following way: on the first visit to a vertex only $Z$ changes by $\pm 1$ equally likely, while on later visits to the same vertex $(X,Y)$ performs a two-dimensional random walk step. We show that this walk is transient thus answering a question of Benjamini, Kozma and Schapira. One important ingredient of the proof is a dispersion result for martingales.; Supposons que (X, Y, Z) soit une marche aléatoire dans Z3 qui se déplace de la façon suivante : à la première visite en un site, seule la coordonnée Z saute de ±1 avec probabilité uniforme, et aux visites suivantes en ce site (X, Y ) effectue un saut dans l’ensemble {(±1, 0), (0, ±1)} avec probabilité uniforme. Nous montrons que cette marche est transiente, répondant ainsi à une question de Benjamini, Kozma et Schapira. Un ingrédient important de la preuve est un résultat de dispersion pour les martingales.

Published
2014
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40. A new approach to quantitative propagation of chaos for drift, diffusion and jump processes

Author

Bernt Wennberg, Stéphane Mischler, Clément Mouhot, CEntre de REcherches en MAthématiques de la DEcision (CEREMADE), Centre National de la Recherche Scientifique (CNRS)-Université Paris Dauphine-PSL, Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL), Department of Pure Mathematics and Mathematical Statistics (DPMMS), Faculty of mathematics Centre for Mathematical Sciences [Cambridge] (CMS), University of Cambridge [UK] (CAM)-University of Cambridge [UK] (CAM), Department of Mathematical Sciences, Chalmers University of Technology [Göteborg]-University of Gothenburg (GU), ANR-08-BLAN-0220,MADCOF,Méthodes Aléatoires et Déterministes pour les processus de COllision, coalescence, de Fragmentation(2008), European Project: 279600,EC:FP7:ERC,ERC-2011-StG_20101014,MATKIT(2011), Université Paris Dauphine-PSL, and Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS)

Subjects
Statistics and Probability, McKean-Vlasov equation, fluctuations, 01 natural sciences, Stability (probability), Boltzmann equation, 010104 statistics & probability, symbols.namesake, Mathematics - Analysis of PDEs, quantitative, FOS: Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Statistical physics, Limit (mathematics), 0101 mathematics, Diffusion (business), inelastic collision, Mathematics, granular gas, 010102 general mathematics, Probability (math.PR), mean-field limit, [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], Nonlinear system, Flow (mathematics), Boltzmann constant, Jump, symbols, drift-diffusion, Statistics, Probability and Uncertainty, Jump process, Analysis, 76P05, 76T25, 60J75, 60J60, Mathematics - Probability, Analysis of PDEs (math.AP)
Abstract

This paper is devoted the the study of the mean field limit for many-particle systems undergoing jump, drift or diffusion processes, as well as combinations of them. The main results are quantitative estimates on the decay of fluctuations around the deterministic limit and of correlations between particles, as the number of particles goes to infinity. To this end we introduce a general functional framework which reduces this question to the one of proving a purely functional estimate on some abstract generator operators (consistency estimate) together with fine stability estimates on the flow of the limiting nonlinear equation (stability estimates). Then we apply this method to a Boltzmann collision jump process (for Maxwell molecules), to a McKean-Vlasov drift-diffusion process and to an inelastic Boltzmann collision jump process with (stochastic) thermal bath. To our knowledge, our approach yields the first such quantitative results for a combination of jump and diffusion processes., v2 (55 pages): many improvements on the presentation, v3: correction of a few typos, to appear In Probability Theory and Related Fields

Published
2013
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41. Chaos and entropic chaos in Kac's model without high moments

Author

Kleber Carrapatoso, Amit Einav, CEntre de REcherches en MAthématiques de la DEcision (CEREMADE), Centre National de la Recherche Scientifique (CNRS)-Université Paris Dauphine-PSL, Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL), Department of Pure Mathematics and Mathematical Statistics (DPMMS), Faculty of mathematics Centre for Mathematical Sciences [Cambridge] (CMS), University of Cambridge [UK] (CAM)-University of Cambridge [UK] (CAM), and Apollo - University of Cambridge Repository

Subjects
Statistics and Probability, 82B40, Kullback–Leibler divergence, Gaussian, Entropy, Chaotic, FOS: Physical sciences, 70F99, Local Levy central limit theorem, 01 natural sciences, 60F05, 70F99, 60J75 and 82B40, Entropic Stability, Combinatorics, symbols.namesake, [MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph], 60F05, FOS: Mathematics, Entropy (information theory), Statistical physics, 0101 mathematics, Mathematical Physics, Mathematics, Central limit theorem, Stable state, Local Lévy central theorem, Probability (math.PR), 010102 general mathematics, Mathematical Physics (math-ph), Entropic Chaos, [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], 010101 applied mathematics, Semi-continuity, Kac's model, symbols, Statistics, Probability and Uncertainty, 60J75, Mathematics - Probability
Abstract

In this paper we present a new local L\'evy Central Limit Theorem, showing convergence to stable states that are not necessarily the Gaussian, and use it to find new and intuitive entropically chaotic families with underlying one-particle function that has moments of order $2\alpha$, with $1, Comment: 42 Pages

Published
2013

42. Exponential Rate of Convergence to Equilibrium for a Model Describing Fiber Lay-Down Processes

Author

Axel Klar, Jean Dolbeault, Christian Schmeiser, Clément Mouhot, CEntre de REcherches en MAthématiques de la DEcision (CEREMADE), Centre National de la Recherche Scientifique (CNRS)-Université Paris Dauphine-PSL, Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL), Departement of Mathematics, TU Kaiserslautern, University of Kaiserslautern, Department of Pure Mathematics and Mathematical Statistics (DPMMS), Faculty of mathematics Centre for Mathematical Sciences [Cambridge] (CMS), University of Cambridge [UK] (CAM)-University of Cambridge [UK] (CAM), Fakultät für Mathematik [Wien], and Universität Wien

Subjects
hypocoercivity, 01 natural sciences, convergence to equilibrium, Mathematics - Analysis of PDEs, spectral gap, FOS: Mathematics, Applied mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], [SPI.GPROC]Engineering Sciences [physics]/Chemical and Process Engineering, exponential rate of convergence, 0101 mathematics, Macro, transport operator, Mathematics, fiber dynamics, Exponential convergence, Applied Mathematics, 010102 general mathematics, Fokker-Planck equation, hypoelliptic operators, kinetic equations, stochastic differential equations, large time behavior, Exponential function, 010101 applied mathematics, Computational Mathematics, Rate of convergence, Poincaré inequality, Norm (mathematics), Diffusion limit, degenerate diffusion, Analysis, Stationary state, Analysis of PDEs (math.AP)
Abstract

This paper is devoted to the adaptation of the method developed in [4,3] to a Fokker-Planck equation for fiber lay-down which has been studied in [1,5]. Exponential convergence towards a unique stationary state is proved in a norm which is equivalent to a weighted $L^2$ norm. The method is based on a micro / macro decomposition which is well adapted to the diffusion limit regime., Comment: 8 pages

Published
2012
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43. The closed knight tour problem in higher dimensions

Author

Bruno Golenia, Joshua Erde, Sylvain Golénia, Computer Science Department [Bristol], University of Bristol [Bristol], Institut de Mathématiques de Bordeaux (IMB), 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), Department of Pure Mathematics and Mathematical Statistics (DPMMS), Faculty of mathematics Centre for Mathematical Sciences [Cambridge] (CMS), and University of Cambridge [UK] (CAM)-University of Cambridge [UK] (CAM)

Subjects
0102 computer and information sciences, 01 natural sciences, Theoretical Computer Science, Combinatorics, symbols.namesake, Extension (metaphysics), Chessboard, [MATH.MATH-CO]Mathematics [math]/Combinatorics [math.CO], FOS: Mathematics, Mathematics - Combinatorics, Discrete Mathematics and Combinatorics, 0101 mathematics, Mathematics, Hamiltonian cycle, Applied Mathematics, Knight's tour, 010102 general mathematics, 16. Peace & justice, Hamiltonian path, Computational Theory and Mathematics, 010201 computation theory & mathematics, symbols, Knight, Geometry and Topology, Combinatorics (math.CO), 05C45,00A08
Abstract

The problem of existence of closed knight tours for rectangular chessboards was solved by Schwenk in 1991. Last year, in 2011, DeMaio and Mathew provide an extension of this result for 3-dimensional rectangular boards. In this article, we give the solution for $n$-dimensional rectangular boards, for $n\geq 4$., Comment: This is a merged version of the previous version and the approach of Erde arXiv:1202.5548

Published
2012

44. Kac's Program in Kinetic Theory

Author

Clément Mouhot, Stéphane Mischler, CEntre de REcherches en MAthématiques de la DEcision (CEREMADE), Centre National de la Recherche Scientifique (CNRS)-Université Paris Dauphine-PSL, Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL), Department of Pure Mathematics and Mathematical Statistics (DPMMS), Faculty of mathematics Centre for Mathematical Sciences [Cambridge] (CMS), and University of Cambridge [UK] (CAM)-University of Cambridge [UK] (CAM)

Subjects
Maxwell molecules, General Mathematics, master equation, jump process, 01 natural sciences, Boltzmann equation, 010104 statistics & probability, Kac's program, Mathematics - Analysis of PDEs, quantitative, collision process, Master equation, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], 0101 mathematics, uniform in time, 82C40, 76P05, 54C70, 60J75, Mathematical physics, Mathematics, Conjecture, 010102 general mathematics, Mathematical statistics, mean-field limit, [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], Bounded function, hard spheres, Kinetic theory of gases, kinetic theory, non cutoff, Jump process, Mathematics - Probability, Identical particles
Abstract

This paper is devoted to the study of propagation of chaos and mean-field limits for systems of indistinguable particles, undergoing collision processes. The prime examples we will consider are the many-particle jump processes of Kac and McKean \cite{Kac1956,McKean1967} giving rise to the Boltzmann equation. We solve the conjecture raised by Kac \cite{Kac1956}, motivating his program, on the rigorous connection between the long-time behavior of a collisional many-particle system and the one of its mean-field limit, for bounded as well as unbounded collision rates. Motivated by the inspirative paper by Gr\"unbaum \cite{Grunbaum}, we develop an abstract method that reduces the question of propagation of chaos to that of proving a purely functional estimate on generator operators ({\em consistency estimates}), along with differentiability estimates on the flow of the nonlinear limit equation ({\em stability estimates}). This allows us to exploit dissipativity at the level of the mean-field limit equation rather than the level of the particle system (as proposed by Kac). Using this method we show: (1) Quantitative estimates, that are uniform in time, on the chaoticity of a family of states. (2) Propagation of {\it entropic chaoticity}, as defined in \cite{CCLLV}. (3) Estimates on the time of relaxation to equilibrium, that are \emph{independent of the number of particles in the system}. Our results cover the two main Boltzmann physical collision processes with unbounded collision rates: hard spheres and \emph{true} Maxwell molecules interactions. The proof of the \emph{stability estimates} for these models requires significant analytic efforts and new estimates., Comment: 101 pages. This preprint supersedes hal-00447988 / arXiv:1001.2994. It entirely includes these previous results and contains new results (relaxation rates independent of the number of particles, propagation of entropic chaos, comparison with approaches based on the BBGKY hierarchy) and some additional explanations and historical notes. This revised version corrects many typos and improves the presentation

Published
2011

45. On Measure Solutions of the Boltzmann Equation, part I: Moment Production and Stability Estimates

Author

Xuguang Lu, Clément Mouhot, Department of Mathematical Sciences, Tsinghua University [Beijing] (THU), Department of Pure Mathematics and Mathematical Statistics (DPMMS), Faculty of mathematics Centre for Mathematical Sciences [Cambridge] (CMS), University of Cambridge [UK] (CAM)-University of Cambridge [UK] (CAM), and Apollo - University of Cambridge Repository

Subjects
Hard spheres, Moment production, POTENTIALS, 01 natural sciences, Stability (probability), Measure (mathematics), Measure solution, ENERGY, Boltzmann equation, 010104 statistics & probability, ANGULAR SINGULARITY, Mathematics - Analysis of PDEs, CONVERGENCE, Hard potentials, FOS: Mathematics, Cutoff, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Uniqueness, 0101 mathematics, Exponential moment, Stability estimate, Spatially homogeneous, Mathematics, Applied Mathematics, 010102 general mathematics, Mathematical analysis, 35Q20, 76P05, Moment (mathematics), SPATIALLY HOMOGENEOUS BOLTZMANN, Long-range interactions, UNIQUENESS, EQUILIBRIUM, INEQUALITIES, Moment estimate, Energy (signal processing), Analysis, Analysis of PDEs (math.AP), Mehler transform
Abstract

The spatially homogeneous Boltzmann equation with hard potentials is considered for measure valued initial data having finite mass and energy. We prove the existence of \emph{weak measure solutions}, with and without angular cutoff on the collision kernel; the proof in particular makes use of an approximation argument based on the Mehler transform. Moment production estimates in the usual form and in the exponential form are obtained for these solutions. Finally for the Grad angular cutoff, we also establish uniqueness and strong stability estimate on these solutions., New version: many typos corrected, and more explanations added. 51 pages, to appear in J. Diff. Eq

Published
2011

46. About Kac's Program in Kinetic Theory

Author

Clement Mouhot Mouhot, Stéphane Mischler, CEntre de REcherches en MAthématiques de la DEcision (CEREMADE), Centre National de la Recherche Scientifique (CNRS)-Université Paris Dauphine-PSL, Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL), Department of Pure Mathematics and Mathematical Statistics (DPMMS), Faculty of mathematics Centre for Mathematical Sciences [Cambridge] (CMS), University of Cambridge [UK] (CAM)-University of Cambridge [UK] (CAM), and Apollo - University of Cambridge Repository

Subjects
propagation of chaos, math-ph, jump process, 01 natural sciences, math.PR, Boltzmann equation, 010104 statistics & probability, symbols.namesake, Mathematics - Analysis of PDEs, math.MP, relaxation, [MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph], Master equation, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], 0101 mathematics, Boltzmann's entropy formula, Mathematical Physics, math.AP, Mathematics, Mathematical physics, Kac, Entropy (statistical thermodynamics), Stochastic process, 010102 general mathematics, Mathematical analysis, General Medicine, Hard spheres, mean-field limit, [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], Boltzmann constant, symbols, Jump process, Mathematics - Probability
Abstract

In this Note we present the main results from the recent work arxiv:1107.3251, which answers several conjectures raised fifty years ago by Kac. There Kac introduced a many-particle stochastic process (now denoted as Kac's master equation) which, for chaotic data, converges to the spatially homogeneous Boltzmann equation. We answer the three following questions raised in \cite{kac}: (1) prove the propagation of chaos for realistic microscopic interactions (i.e. in our results: hard spheres and true Maxwell molecules); (2) relate the time scales of relaxation of the stochastic process and of the limit equation by obtaining rates independent of the number of particles; (3) prove the convergence of the many-particle entropy towards the Boltzmann entropy of the solution to the limit equation (microscopic justification of the $H$-theorem of Boltzmann in this context). These results crucially rely on a new theory of quantitative uniform in time estimates of propagation of chaos., Comment: 7 pages. Announcement of the results in arXiv:1107.3251. To appear in Comptes Rendus Acad. Sciences Paris

Published
2011

47. On Landau damping

Author

Cédric Villani, Clément Mouhot, Reymond, Aurélie, Département de Mathématiques et Applications - ENS Paris (DMA), École normale supérieure - Paris (ENS-PSL), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS), Department of Applied Mathematics and Theoretical Physics (DAMTP), University of Cambridge [UK] (CAM), Unité de Mathématiques Pures et Appliquées (UMPA-ENSL), École normale supérieure de Lyon (ENS de Lyon)-Centre National de la Recherche Scientifique (CNRS), Department of Pure Mathematics and Mathematical Statistics (DPMMS), Faculty of mathematics Centre for Mathematical Sciences [Cambridge] (CMS), University of Cambridge [UK] (CAM)-University of Cambridge [UK] (CAM), 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-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 normale supérieure - Paris (ENS Paris), Centre National de la Recherche Scientifique (CNRS)-École normale supérieure - Lyon (ENS Lyon), Centre National de la Recherche Scientifique (CNRS)-École normale supérieure - Paris (ENS Paris), and Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)

Subjects
General Mathematics, 01 natural sciences, Stability (probability), Landau damping, 010305 fluids & plasmas, Mathematics - Analysis of PDEs, [PHYS.PHYS.PHYS-PLASM-PH]Physics [physics]/Physics [physics]/Plasma Physics [physics.plasm-ph], 0103 physical sciences, Coulomb, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Limit (mathematics), 0101 mathematics, [PHYS.COND.CM-SM]Physics [physics]/Condensed Matter [cond-mat]/Statistical Mechanics [cond-mat.stat-mech], Condensed Matter - Statistical Mechanics, ComputingMilieux_MISCELLANEOUS, Mathematics, Kolmogorov–Arnold–Moser theorem, plasma physics, 010102 general mathematics, Mathematical analysis, Vlasov equation, galactic dynamics, 16. Peace & justice, Astrophysics - Astrophysics of Galaxies, Physics - Plasma Physics, Exponential function, [PHYS.ASTR.GA]Physics [physics]/Astrophysics [astro-ph]/Galactic Astrophysics [astro-ph.GA], 35B35 (35J05, 62E20, 70F15, 82C99, 85A05, 82D10, 35Q60, 76X05), [SDU.ASTR.GA]Sciences of the Universe [physics]/Astrophysics [astro-ph]/Galactic Astrophysics [astro-ph.GA], Convection–diffusion equation, Vlasov-Poisson equation
Abstract

Going beyond the linearized study has been a longstanding problem in the theory of Landau damping. In this paper we establish exponential Landau damping in analytic regularity. The damping phenomenon is reinterpreted in terms of transfer of regularity between kinetic and spatial variables, rather than exchanges of energy; phase mixing is the driving mechanism. The analysis involves new families of analytic norms, measuring regularity by comparison with solutions of the free transport equation; new functional inequalities; a control of nonlinear echoes; sharp scattering estimates; and a Newton approximation scheme. Our results hold for any potential no more singular than Coulomb or Newton interaction; the limit cases are included with specific technical effort. As a side result, the stability of homogeneous equilibria of the nonlinear Vlasov equation is established under sharp assumptions. We point out the strong analogy with the KAM theory, and discuss physical implications., Comment: News: (1) the main result now covers Coulomb and Newton potentials, and (2) some classes of Gevrey data; (3) as a corollary this implies new results of stability of homogeneous nonmonotone equilibria for the gravitational Vlasov-Poisson equation

Published
2011

48. On the Integrability of Tonelli Hamiltonians

Author

Alfonso Sorrentino, Sorrentino, Alfonso, CEntre de REcherches en MAthématiques de la DEcision (CEREMADE), Centre National de la Recherche Scientifique (CNRS)-Université Paris Dauphine-PSL, Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL), Department of Pure Mathematics and Mathematical Statistics (DPMMS), Faculty of mathematics Centre for Mathematical Sciences [Cambridge] (CMS), University of Cambridge [UK] (CAM)-University of Cambridge [UK] (CAM), and ANR-07-BLAN-0361,KAMFAIBLE,Hamilton-Jacobi et théorie KAM faible : à l'interface des EDP, systèmes dynamiques lagrangiens et symboliques(2007)

Subjects
Computer Science::Machine Learning, Mathematics::Dynamical Systems, General Mathematics, Tonelli Hamiltonians, [MATH.MATH-DS]Mathematics [math]/Dynamical Systems [math.DS], FOS: Physical sciences, Integrals of motion, Dynamical Systems (math.DS), commuting Hamiltonians, Computer Science::Digital Libraries, Hamiltonian system, Statistics::Machine Learning, symbols.namesake, MSC: 37J50, 37J35, Settore MAT/05 - Analisi Matematica, FOS: Mathematics, Mathematics (all), Mathematics - Dynamical Systems, Mathematics::Symplectic Geometry, Invariant Lagrangian foliations, Mathematical Physics, Mathematical physics, Mathematics, Applied Mathematics, Torus, Mathematical Physics (math-ph), 37J50, 37J35, Mathematics - Symplectic Geometry, Computer Science::Mathematical Software, symbols, Symplectic Geometry (math.SG), Aubry-Mather theory, Hamiltonian (quantum mechanics), Liouville theorem
Abstract

In this article we discuss a weaker version of Liouville's theorem on the integrability of Hamiltonian systems. We show that in the case of Tonelli Hamiltonians the involution hypothesis on the integrals of motion can be completely dropped and still interesting information on the dynamics of the system can be deduced. Moreover, we prove that on the n-dimensional torus this weaker condition implies classical integrability in the sense of Liouville. The main idea of the proof consists in relating the existence of independent integrals of motion of a Tonelli Hamiltonian to the size of its Mather and Aubry sets. As a byproduct we point out the existence of non-trivial common invariant sets for all Hamiltonians that Poisson-commute with a Tonelli one., 19 pages. Version accepted by Trans. Amer. Math. Soc

Published
2011

49. Survival of near-critical branching Brownian motion

Author

Jason Schweinsberg, Nathanaël Berestycki, Julien Berestycki, 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), Department of Pure Mathematics and Mathematical Statistics (DPMMS), Faculty of mathematics Centre for Mathematical Sciences [Cambridge] (CMS), University of Cambridge [UK] (CAM)-University of Cambridge [UK] (CAM), Department of Mathematics at the University of California at San Diego, University of California [San Diego] (UC San Diego), University of California-University of California, ANR-08-BLAN-0220,MADCOF,Méthodes Aléatoires et Déterministes pour les processus de COllision, coalescence, de Fragmentation(2008), Department of Mathematics [Univ California San Diego] (MATH - UC San Diego), and University of California (UC)-University of California (UC)

Subjects
Near critical, Probability (math.PR), 010102 general mathematics, Zero (complex analysis), Statistical and Nonlinear Physics, 01 natural sciences, [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], 010104 statistics & probability, Survival probability, FOS: Mathematics, Traveling wave, 60J99 (Primary), 60J80, 60F17, 60G15 (Secondary), 0101 mathematics, ComputingMilieux_MISCELLANEOUS, Mathematical Physics, Brownian motion, Positive probability, Mathematics - Probability, Mathematics, Mathematical physics
Abstract

International audience; Consider a system of particles performing branching Brownian motion with negative drift mu = root 2-epsilon and killed upon hitting zero. Initially there is one particle at x > 0. Kesten (Stoch. Process. Appl. 7: 9-47, 1978) showed that the process survives with positive probability if and only if e > 0. Here we are interested in the asymptotics as e. 0 of the survival probability Q(mu)(x). It is proved that if L = p/v e then for all x. R, lim(epsilon -> 0) Q(mu)(L + x) = theta(x) is an element of (0, 1) exists and is a traveling wave solution of the Fisher-KPP equation. Furthermore, we obtain sharp asymptotics of the survival probability when x < L and L - x -> infinity. The proofs rely on probabilistic methods developed by the authors in (Berestycki et al. in arXiv:1001.2337, 2010). This completes earlier work by Harris, Harris and Kyprianou (Ann. Inst. Henri Poincare Probab. Stat. 42: 125-145, 2006) and confirms predictions made by Derrida and Simon (Europhys. Lett. 78: 60006, 2007), which were obtained using nonrigorous PDE methods.

Published
2010

50. The structure of popular difference sets

Author

Julia Wolf, Department of Pure Mathematics and Mathematical Statistics (DPMMS), Faculty of mathematics Centre for Mathematical Sciences [Cambridge] (CMS), University of Cambridge [UK] (CAM)-University of Cambridge [UK] (CAM), Institute for Advanced Study, Princeton University, and Juppin, Carole

Subjects
Discrete mathematics, Difference set, General Mathematics, 010102 general mathematics, Structure (category theory), Sumset, 01 natural sciences, Measure (mathematics), Set (abstract data type), 010104 statistics & probability, Total variation, Large set (Ramsey theory), Arithmetic progression, 0101 mathematics, ComputingMilieux_MISCELLANEOUS, Mathematics
Abstract

We show that the set of popular differences of a large subset of ℤN does not always contain the complete difference set of another large set. For this purpose we construct a so-called niveau set, which was first introduced by Ruzsa in [Ruz87] and later used in [Ruz91] to show that there exists a large subset of ℤN whose sumset does not contain any long arithmetic progressions. In this paper we make substantial use of measure concentration results on the multi-dimensional torus and Esseen’s Inequality.

Published
2010

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