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1. Communicating the intangible: an Anishnaabeg story

Author

Pitawanakwat, Kenn and Paper, Jordan

Subjects
Ojibwas -- Religious aspects -- Analysis, Ethnomethodology -- Analysis -- Religious aspects, Anthropology/archeology/folklore, Ethnic, cultural, racial issues/studies, History, Literature/writing, Analysis, Religious aspects
Abstract

In the enterprise of comparative religion, as with crosscultural studies in general, a major methodological problem centers on communication: How can the understanding of one culture be expressed in such [...]

Published
1996

2. Stability of optimal traffic plans in the irrigation problem

Author

Antoine Prouff, Antonio De Rosa, Andrea Marchese, Maria Colombo, Paul Pegon, Ecole Polytechnique Fédérale de Lausanne (EPFL), Courant Institute of Mathematical Sciences [New York] (CIMS), New York University [New York] (NYU), NYU System (NYU)-NYU System (NYU), Department of mathematics/Dipartimento di Matematica [Univ. Trento], Università degli Studi di Trento (UNITN), 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), 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)-Inria de Paris, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Ecole Normale Supérieure Paris-Saclay (ENS Paris Saclay), Maria Colombo was partially supported by the Swiss National Science Foundation grant 200021_182565. Antonio De Rosa has been supported by the NSF DMS Grant No. 1906451. Andrea Marchese acknowledges partial support from GNAMPA-INdAM., Department of Mathematics [College Park], University of Maryland [College Park], University of Maryland System-University of Maryland System, Bocconi Institute for Data Science and Analytics (BIDSA), Bocconi University [Milan, Italy], 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 sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS)-Inria de Paris, M. C. was partially supported by the Swiss National Science Foundation grant 200021_182565. A. D. R. has been supported by the NSF DMS Grant No. 1906451 and the NSF DMS Grant No. 2112311. A. M. acknowledges partial support from GNAMPA-INdAM. A.P. was supported by the Fondation Mathématiques Jacques Hadamard. P.P and A.P. both acknowledge EPFL for hosting them during the semester this paper was prepared., Inria de Paris, and 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)

Subjects
BRANCHED TRANSPORT, Mathematical optimization, Irrigation, TRAFFIC PLANS, [MATH.MATH-CA]Mathematics [math]/Classical Analysis and ODEs [math.CA], 01 natural sciences, Stability (probability), symbols.namesake, Mathematics - Analysis of PDEs, Classical Analysis and ODEs (math.CA), FOS: Mathematics, Computer Science::Networking and Internet Architecture, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Discrete Mathematics and Combinatorics, Limit (mathematics), 0101 mathematics, Mathematics - Optimization and Control, Mathematics, TRANSPORTATION NETWORK, BRANCHED TRANSPORT, IRRIGATION PROBLEM, TRAFFIC PLANS, STABILITY, STABILITY, IRRIGATION PROBLEM, Applied Mathematics, TRANSPORTATION NETWORK, 010102 general mathematics, Eulerian path, Flow network, 010101 applied mathematics, Optimization and Control (math.OC), Mathematics - Classical Analysis and ODEs, symbols, [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], Analysis, Lagrangian, Analysis of PDEs (math.AP)
Abstract

We prove the stability of optimal traffic plans in branched transport. In particular, we show that any limit of optimal traffic plans is optimal as well. This result goes beyond the Eulerian stability proved in [7], extending it to the Lagrangian framework.

Published
2022

3. Rigorous study of the equilibria of collision kernels appearing in the theory of weak turbulence

Author

Maxime Breden, Laurent Desvillettes, Technische Universität Munchen - Université Technique de Munich [Munich, Allemagne] (TUM), Université Paris Diderot, Sorbonne Paris Cité, Paris, France, Université Paris Diderot - Paris 7 (UPD7), and The research leading to this paper was partly funded by Université SorbonneParis Cité, in the framework of the 'Investissements d'Avenir', convention ANR-11-IDEX-0005. MB also acknowledges partial support from a Lichtenberg Professorship grant ofthe VolkswagenStiftung awarded to C. Kuehn.

Subjects
Keys words: weak turbulence, Complex system, 82C40, FOS: Physical sciences, Type (model theory), [MATH.MATH-FA]Mathematics [math]/Functional Analysis [math.FA], 01 natural sciences, symbols.namesake, equilibria Mathematical Subject Classification: 76F99, Mathematics - Analysis of PDEs, Mathematics (miscellaneous), [MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph], FOS: Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Statistical physics, 0101 mathematics, Mathematical Physics, Physics, 76P05, Turbulence, Mechanical Engineering, 010102 general mathematics, Mathematical Subject Classification: 76F99, 76P05, 82C40, Turbulence theory, Mathematical Physics (math-ph), Collision, equilibria, Functional Analysis (math.FA), Mathematics - Functional Analysis, 010101 applied mathematics, collision kernels, Boltzmann constant, Boltzmann's H-theorem, symbols, Particle, weak turbulence, Analysis, Analysis of PDEs (math.AP)
Abstract

In this paper, we rigorously obtain all the equilibria of collision kernels of type ``two particles give two particles'' appearing in weak turbulence theory under very general assumptions, thus completing the ``equality case'' in Boltzmann's H-theorem for those models. We also provide some rigorous results for collision kernels of type ``two particles give one particle'', under assumptions which include some of the most classical kernels of this type. The method of proof is inspired by the quantitative estimates obtained for the Landau equation by Desvillettes in [J. Funct. Anal. 269:1359-1403, 2015].

Published
2018

4. On discretization schemes for stochastic evolution equations

Author

Annie Millet, Istvan Gyongy, School of Mathematics - University of Edinburgh, University of Edinburgh, 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), Statistique Appliquée et MOdélisation Stochastique (SAMOS), Université Paris 1 Panthéon-Sorbonne (UP1), Modélisation Appliquée, Trajectoires Institutionnelles et Stratégies Socio-Économiques (MATISSE - UMR 8595), Université Paris 1 Panthéon-Sorbonne (UP1)-Centre National de la Recherche Scientifique (CNRS), This paper was written while I. Gyongy was visiting the University of Paris X. The research of I. Gyongy is partially supported byEU Network HARP. The research of A. Millet is partially supported by the research project BMF2003-01345, and Centre National de la Recherche Scientifique (CNRS)-Université Paris Diderot - Paris 7 (UPD7)-Université Pierre et Marie Curie - Paris 6 (UPMC)

Subjects
Discretization, 010102 general mathematics, Probability (math.PR), Banach space, MathematicsofComputing_NUMERICALANALYSIS, Stochastic evolution, 01 natural sciences, 010101 applied mathematics, [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], Monotone polygon, Convergence (routing), Monotone operators, FOS: Mathematics, Applied mathematics, finite elements, 60H15 65M60, coercivity, 0101 mathematics, Analysis, Stochastic evolution equations, Mathematics - Probability, Mathematics
Abstract

International audience; Stochastic evolutional equations with monotone operators are considered in Banach spaces. Explicit and implicit numerical schemes are presented. The convergence of the approximations to the solution of the equations is proved.

Published
2006

5. From left modules to algebras over an operad: application to combinatorial Hopf algebras

Author

Muriel Livernet, Laboratoire Analyse, Géométrie et Applications (LAGA), Université Paris 8 Vincennes-Saint-Denis (UP8)-Centre National de la Recherche Scientifique (CNRS)-Institut Galilée-Université Paris 13 (UP13), and this paper was written during the stay of the author at Institut Mittag-Leffler (djursholm, sweden).

Subjects
Pure mathematics, twisted bialgebra, 18D50 (Primary), 16W30, 16A06 (Secondary), Context (language use), 0102 computer and information sciences, Quasitriangular Hopf algebra, 01 natural sciences, Mathematics::Algebraic Topology, Mathematics::Category Theory, Mathematics::Quantum Algebra, FOS: Mathematics, S-module, 0101 mathematics, Mathematics, operad, Algebra and Number Theory, Functor, Quantum group, Applied Mathematics, 010102 general mathematics, [MATH.MATH-RA]Mathematics [math]/Rings and Algebras [math.RA], Mathematics::Rings and Algebras, Tensor algebra, combinatorial Hopf algebra, Mathematics - Rings and Algebras, Hopf algebra, MSC2000: Primary: 18D50, Secondary: 16W30, 16A06, Rings and Algebras (math.RA), 010201 computation theory & mathematics, free associative algebra, Geometry and Topology, Forgetful functor, Analysis, Vector space
Abstract

The purpose of this paper is two fold: we study the behaviour of the forgetful functor from S-modules to graded vector spaces in the context of algebras over an operad and derive from this theory the construction of combinatorial Hopf algebras. As a byproduct we obtain freeness and cofreeness results for these Hopf algebras. Let O denote the forgetful functor from S-modules to graded vector spaces. Left modules over an operad P are treated as P-algebras in the category of S-modules. We generalize the results obtained by Patras and Reutenauer in the associative case to any operad P: the functor O sends P-algebras to P-algebras. If P is a Hopf operad then O sends Hopf P-algebras to Hopf P-algebras. If the operad P is regular one gets two different structures of Hopf P-algebras in the category of graded vector spaces. We develop the notion of unital infinitesimal P-bialgebra and prove freeness and cofreeness results for Hopf algebras built from Hopf operads. Finally, we prove that many combinatorial Hopf algebras arise from our theory, as Hopf algebras on the faces of the permutohedra and associahedra., Section 4.3 removed. To appear in Annales Math\'ematiques Blaise Pascal

Published
2006

6. A Theory of L 1-Dissipative Solvers for Scalar Conservation Laws with Discontinuous Flux

Author

Kenneth H. Karlsen, Boris Andreianov, Nils Henrik Risebro, Laboratoire de Mathématiques de Besançon (UMR 6623) (LMB), Université de Bourgogne (UB)-Université de Franche-Comté (UFC), Université Bourgogne Franche-Comté [COMUE] (UBFC)-Université Bourgogne Franche-Comté [COMUE] (UBFC)-Centre National de la Recherche Scientifique (CNRS), Center of Mathematics for Applications [Oslo] (CMA), Department of Mathematics [Oslo], Faculty of Mathematics and Natural Sciences [Oslo], University of Oslo (UiO)-University of Oslo (UiO)-Faculty of Mathematics and Natural Sciences [Oslo], University of Oslo (UiO)-University of Oslo (UiO), and This paper was written as part of the research program on Nonlinear Partial Dierential Equations at the Centre for Advanced Study at the Norwegian Academy of Science and Letters in Oslo, which took place during the academic year 2008-09.

Subjects
Pure mathematics, Admissibility, Adapted entropy, 01 natural sciences, Mathematics - Analysis of PDEs, Mathematics (miscellaneous), Boundary trace, FOS: Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Discontinuous flux, Germ, Uniqueness, Mathematics - Numerical Analysis, 0101 mathematics, Scalar conservation law, Mathematics, Conservation law, Finite volume method, Mechanical Engineering, 010102 general mathematics, Primary 35L65, Secondary 35R05, Numerical Analysis (math.NA), 010101 applied mathematics, Convergence of numerical approximations, Uniqueness criteria, Finite volume scheme, Piecewise, Dissipative system, Adapted viscosity, Entropy solution, Vanishing viscosity, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA], Analysis, Analysis of PDEs (math.AP)
Abstract

International audience; We propose a general framework for the study of L1 contractive semigroups of solutions to conservation laws with discontinuous flux: (CL) u_t + f(x; u)_x = 0; f(x; u) = f^l(u), x < 0 and f(x;u)= f^r(u); x > 0; where the fluxes f^l and f^r are mainly assumed to be continuous. Developing the ideas of a number of preceding works (Baiti and Jenssen [14], Audusse and Perthame [12], Garavello et al. [35], Adimurthi et al. [3], Buerger et al. [21]), we claim that the whole admissibility issue is reduced to the selection of a family of "elementary solutions", which are piecewise constant weak solutions of the form c(x) = c^l 1l_{x0}. We refer to such a family as a "germ". It is well known that (CL) admits many different L1 contractive semigroups, some of which reflects different physical applications. We revisit a number of the existing admissibility (or entropy) conditions and identify the germs that underly these conditions. We devote specic attention to the "vanishing viscosity" germ, which is a way to express the "Gamma-condition" of Diehl [32]. For any given germ, we formulate "germ-based" admissibility conditions in the form of a trace condition on the flux discontinuity line {x = 0} (in the spirit of Vol'pert [80]) and in the form of a family of global entropy inequalities (following Kruzhkov [50] and Carrillo [22]). We characterize those germs that lead to the L1-contraction property for the associated admissible solutions. Our approach offers a streamlined and unifying perspective on many of the known entropy conditions, making it possible to recover earlier uniqueness results under weaker conditions than before, and to provide new results for other less studied problems. Several strategies for proving the existence of admissible solutions are discussed, and existence results are given for fluxes satisfying some additional conditions. These are based on convergence results either for the vanishing viscosity method (with standard viscosity or with specic viscosities "adapted" to the choice of a germ), or for specific germ-adapted finite volume schemes.

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