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151. Wick polynomials in non-commutative probability: A group-theoretical approach

Author

Ebrahimi-Fard, Kurusch, Patras, Frédéric, Tapia, Nikolas, and Zambotti, Lorenzo

Subjects
Wick polynomials, free cumulants, Nuclear Theory, combinatorial Hopf algebra, General Relativity and Quantum Cosmology, boolean cumulants, group actions, 16T05, Mathematics::Probability, Mathematics::Quantum Algebra, monotone cumulants, formal power series, Mathematics::Mathematical Physics, 16T30, shuffle algebra, 16T10
Abstract

Wick polynomials and Wick products are studied in the context of non-commutative probability theory. It is shown that free, boolean and conditionally free Wick polynomials can be defined and related through the action of the group of characters over a particular Hopf algebra. These results generalize our previous developments of a Hopf algebraic approach to cumulants and Wick products in classical probability theory.

Published
2020
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156. Épistémologie historique et architecture du savoir

Author

Cantu, Paola, Patras, Frédéric, Centre National de la Recherche Scientifique (CNRS), Aix Marseille Université (AMU), Centre Gilles-Gaston Granger (CGGG), Centre National de la Recherche Scientifique (CNRS)-Aix Marseille Université (AMU), Laboratoire Jean Alexandre Dieudonné (JAD), Université Nice Sophia Antipolis (... - 2019) (UNS), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA), Sébastien Maronne, Institut de Mathématiques de Toulouse UMR 5219, Université Toulouse III Paul Sabatier, Aix Marseille Université (AMU)-Centre National de la Recherche Scientifique (CNRS), and Université Nice Sophia Antipolis (1965 - 2019) (UNS)

Subjects
[SHS.HISPHILSO]Humanities and Social Sciences/History, Philosophy and Sociology of Sciences, [MATH.MATH-HO]Mathematics [math]/History and Overview [math.HO], [SHS.PHIL]Humanities and Social Sciences/Philosophy, ComputingMilieux_MISCELLANEOUS
Abstract

International audience

Published
2019

162. Hopf-algebraic Deformations of Products and Wick Polynomials.

Author

Ebrahimi-Fard, Kurusch, Patras, Frédéric, Tapia, Nikolas, and Zambotti, Lorenzo

Subjects
*POLYNOMIALS, *HOPF algebras, *RANDOM variables, *STRUCTURAL analysis (Engineering), *FUNCTIONALS
Abstract

We present an approach to cumulant–moment relations and Wick polynomials based on extensive use of convolution products of linear functionals on a coalgebra. This allows, in particular, to understand the construction of Wick polynomials as the result of a Hopf algebra deformation under the action of linear automorphisms induced by multivariate moments associated to an arbitrary family of random variables with moments of all orders. We also generalize the notion of deformed product in order to discuss how these ideas appear in the recent theory of regularity structures. [ABSTRACT FROM AUTHOR]

Published
2020
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166. Approches phénoménologiques de la vérité mathématique

Author

Patras, Frédéric, Laboratoire Jean Alexandre Dieudonné (JAD), Université Nice Sophia Antipolis (... - 2019) (UNS), and COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS)

Subjects
[MATH.MATH-RA]Mathematics [math]/Rings and Algebras [math.RA]
Abstract

A paraître; International audience; Ce sera le propos de cet article que d’essayer d’articuler le questionnement et les th`eses heideggériennes sur la vérité aux théories plus classiques etsurtout de mettre en évidence ce qu’ils peuvent apporter aux débats sur la vérité et les fondements des mathématiques.

Published
2016

167. Deformations of shuffles and quasi-shuffles

Author

Foissy, Loïc, Patras, Frédéric, Thibon, Jean-Yves, Laboratoire de Mathématiques Pures et Appliquées Joseph Liouville (LMPA), Université du Littoral Côte d'Opale (ULCO), Laboratoire Jean Alexandre Dieudonné (JAD), Université Nice Sophia Antipolis (1965 - 2019) (UNS), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS), Laboratoire d'Informatique Gaspard-Monge (LIGM), Université Paris-Est Marne-la-Vallée (UPEM)-École des Ponts ParisTech (ENPC)-ESIEE Paris-Fédération de Recherche Bézout-Centre National de la Recherche Scientifique (CNRS), Université Nice Sophia Antipolis (... - 2019) (UNS), and Centre National de la Recherche Scientifique (CNRS)-Fédération de Recherche Bézout-ESIEE Paris-École des Ponts ParisTech (ENPC)-Université Paris-Est Marne-la-Vallée (UPEM)

Subjects
Hausdorff series, Algèbre shuffle, 05E05, 16T30, Mathematics::Category Theory, [MATH.MATH-RA]Mathematics [math]/Rings and Algebras [math.RA], [MATH.MATH-CO]Mathematics [math]/Combinatorics [math.CO], Combinatorial Hopf algebras, Shuffle algebras, algèbre de Hopf combinatoire, série de Hausdorff
Abstract

International audience; We investigate deformations of the shuffle Hopf algebra structure Sh(A) which can be defined on the tensor algebra over a commutative algebra A. Such deformations, leading for example to the quasi-shuffle algebra QSh(A), can be interpreted as natural transformations of the functor Sh, regarded as a functor from commutative nonunital algebras to coalgebras. We prove that the monoid of natural endomophisms of the functor Sh is isomorphic to the monoid of formal power series in one variable without constant term under composition, so that in particular, its natural automorphisms are in bijection with formal diffeomorphisms of the line. These transformations can be interpreted as elements of the Hopf algebra of word quasi-symmetric functions WQSym, and in turn define deformations of its structure. This leads to a new embedding of free quasi-symmetric functions into WQSym, whose relevance is illustrated by a simple and transparent proof of Goldberg's formula for the coefficients of the Hausdorff series.; On s’intéresse aux déformations de la structure d’algèbre de Hopf des battages (ou shuffles) ${\rm Sh}(A)$ définie sur l’algèbre tensorielle sur une algèbre commutative $A$. Ces déformations, dont un cas remarquable est donné par l’algèbre de Hopf des quasi-shuffles ${\rm QSh}(A)$, s’interprètent comme transformations naturelles du foncteur ${\rm Sh}$ vu comme foncteur des algèbres commutatives non unitaires vers les coalgèbres. On montre en particulier que le monoïde des endomorphismes naturels du foncteur ${\rm Sh}$ est isomorphe au monoïde des séries formelles en une variable sans terme constant pour la loi de composition des séries. Les automorphismes naturels du foncteur sont donc en bijection avec les difféomorphismes formels de la droite.Ces transformations s’interprètent aussi comme des élements de l’algèbre de Hopf des surjections (ou, de façon équivalente des fonctions quasi-symétriques en mots) $\mathbf{WQSym}$, et en définissent à leur tour des déformations. Cette remarque conduit entre autres à un nouveau plongement des fonctions quasi-symétriques libres dans $\mathbf{WQSym}$ dont la pertinence est illustrée par une preuve simple de la formule de Goldberg pour les coefficients de la série de Hausdorff.

Published
2016

168. The orthogonal group action on spatial vectors: invariants, covariants, and syzygies

Author

Dhont, Guillaume, Cassam-Chenaï, Patrick, Zhilinskii, Boris, Patras, Frédéric, Laboratoire de Physico-Chimie de l'Atmosphère (LPCA), Université du Littoral Côte d'Opale (ULCO)-Centre National de la Recherche Scientifique (CNRS), Laboratoire Jean Alexandre Dieudonné (JAD), Université Nice Sophia Antipolis (... - 2019) (UNS), and COMUE Université Côte d'Azur (2015 - 2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015 - 2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS)

Subjects
[MATH.MATH-AC]Mathematics [math]/Commutative Algebra [math.AC], [PHYS.MPHY]Physics [physics]/Mathematical Physics [math-ph], PACS: 02.20.-a, 02.20.Hj, 31.15.xh, 31.30.jy, 33.20.Vq
Abstract

The present paper on the SO(3) invariants and covariants built from N vectors of the three–dimensional space is the follow-up of our previous article [1] dealing with planar vectors and SO(2) symmetry. The goal is to propose integrity basis for the set of SO(3) invariants and covariant free modules and easy-to-use generating families in the case of non-free covariants modules. The existence of such non-free modules is one of the noteworthy features unseen when dealing with finite point groups, that we want to point out. As in paper [1], the Molien function plays a central role in the conception of these bases. The Molien functions are computed and checked by the use of two independent paths. The first computation relies on the Molien integral [2] and requires the matrix representation of the group action on the N spatial vectors. The second path considers the Molien function for only one spatial vector as the elementary building material from which are worked out the other Molien functions.

Published
2015

170. Il y a déjà de l'algèbre chez Euclide... ou presque

Author

Patras, Frédéric, Laboratoire Jean Alexandre Dieudonné (JAD), Université Nice Sophia Antipolis (... - 2019) (UNS), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS), and J. Benoist, Th. Paul

Subjects
[MATH.MATH-RA]Mathematics [math]/Rings and Algebras [math.RA], ComputingMilieux_MISCELLANEOUS
Abstract

International audience

Published
2015

171. ${\epsilon}$-Noncrossing Partitions and Cumulants in Free Probability.

Author

Ebrahimi-Fard, Kurusch, Patras, Frédéric, and Speicher, Roland

Subjects
*PARTITIONS (Mathematics), *CUMULANTS, *FREE probability theory, *NONCOMMUTATIVE function spaces, *PERMUTATIONS
Abstract

Motivated by recent work on mixtures of classical and free probabilities, we introduce and study the notion of |$\epsilon$| -noncrossing partitions. It is shown that the set of such partitions forms a lattice. It is a subposet of the poset of partitions and it contains the one of noncrossing partitions. Moreover, |$\epsilon$| -cumulants are introduced and shown to characterize the notion of |$\epsilon$| -independence. [ABSTRACT FROM AUTHOR]

Published
2018
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189. A Mean field theory of nonlinear filtering

Author

del Moral, Pierre, Patras, Frédéric, Rubenthaler, Sylvain, Dan Crisan, Boris L. Rozovskii, Advanced Learning Evolutionary Algorithms (ALEA), Centre National de la Recherche Scientifique (CNRS)-Université de Bordeaux (UB)-Inria Bordeaux - Sud-Ouest, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), 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), Laboratoire Jean Alexandre Dieudonné (JAD), Université Nice Sophia Antipolis (... - 2019) (UNS), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS), 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), 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), and Université Nice Sophia Antipolis (1965 - 2019) (UNS)

Subjects
interacting particle systems, propagations of chaos, Feynman-Kac measures, [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], combinatorial enumeration, nonlinear filtering, historical and genealogical tree models, trees and forests, central limit theorems, Gaussian fields
Abstract

International audience; We present a mean field particle theory for the numerical approximation of Feynman-Kac path integrals in the context of nonlinear filtering. We show that the conditional distribution of the signal paths given a series of noisy and partial observation data is approximated by the occupation measure of a genealogical tree model associated with mean field interacting particle model. The complete historical model converges to the McKean distribution of the paths of a nonlinear Markov chain dictated by the mean field interpretation model. We review the stability properties and the asymptotic analysis of these interacting processes, including fluctuation theorems and large deviation principles. We also present an original Laurent type and algebraic tree-based integral representations of particle block distributions. These sharp and non asymptotic propagations of chaos properties seem to be the first result of this type for mean field and interacting particle systems.

Published
2011

193. Interacting path systems for credit portfolios risk analysis

Author

del Moral, Pierre, Patras, Frédéric, 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), Laboratoire Jean Alexandre Dieudonné (JAD), Université Nice Sophia Antipolis (... - 2019) (UNS), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS), INRIA, and Université Nice Sophia Antipolis (1965 - 2019) (UNS)

Subjects
rare event simulation, interacting particle systems, [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], genetic algorithms, stochastic particle methods, Credit portfolios risk analysis, ACM: G.: Mathematics of Computing, ACM: I.: Computing Methodologies
Abstract

This Note introduces an algorithm (referred to as interacting path systems algorithm, IPaS) based on the first Author multilevel splitting technique and suited to the analysis of multiple defaults in credit portfolios. A full development of this Note incorporating technical details and a survey of the use of Interacting Particle Systems in the field of credit risk, \it Interacting path systems for credit risk, is submitted for publication in \it Recent Advancements in the Theory and Practice of Credit Derivatives, Eds T. Bielecki, D. Brigo, F. Patras, Bloomberg Press (2011). The reader is referred to this article for further details.

Published
2010

194. Modélisation, simulation et analyse de propriétés de réseaux orbitèles

Author

Josselin, Didier, primary, Chekir, Sonia, additional, Pasquet, Alain, additional, Labatut, Vincent, additional, Capowiez, Yvan, additional, Mazzia, Christophe, additional, Hayel, Yezekael, additional, Lammoglia, Adrien, additional, Genre-Grandpierre, Cyrille, additional, Mitsche, Dieter, additional, and Patras, Frédéric, additional

Published
2015
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198. Coalescent tree based functional representations for some Feynman-Kac particle models

Author

del Moral, Pierre, Patras, Frédéric, Rubenthaler, Sylvain, 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), Laboratoire Jean Alexandre Dieudonné (JAD), Université Nice Sophia Antipolis (... - 2019) (UNS), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS), 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), and Université Nice Sophia Antipolis (1965 - 2019) (UNS)

Subjects
65C05, 60J80, 47D08, Probability (math.PR), 92D25, 31B10, [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], 60K35, FOS: Mathematics, 60C05, 65C35, Mathematics - Probability, ComputingMilieux_MISCELLANEOUS
Abstract

We design a theoretic tree-based functional representation of a class of Feynman-Kac particle distributions, including an extension of the Wick product formula to interacting particle systems. These weak expansions rely on an original combinatorial, and permutation group analysis of a special class of forests. They provide refined non asymptotic propagation of chaos type properties, as well as sharp $\LL\_p$-mean error bounds, and laws of large numbers for $U$-statistics. Applications to particle interpretations of the top eigenvalues, and the ground states of Schr\"{o}dinger semigroups are also discussed.

Published
2009

199. Convergence of U-statistics for interacting particle systems

Author

del Moral, Pierre, Patras, Frédéric, Rubenthaler, Sylvain, 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), 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), Laboratoire Jean Alexandre Dieudonné (JAD), Université Nice Sophia Antipolis (... - 2019) (UNS), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS), INRIA, ANR-06-SETI-0009,NEBIANNO,Sécurité et fiabilité des techniques de tatouages(2006), 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), and Université Nice Sophia Antipolis (1965 - 2019) (UNS)

Subjects
[MATH.MATH-PR]Mathematics [math]/Probability [math.PR]
Abstract

The convergence of U-statistics has been intensively studied for estimators based on families of i.i.d. random variables and variants of them. In most cases, the independence assumption is crucial. When dealing with Feynman-Kac and other interacting particle systems of Monte Carlo type, one faces a new type of problem. Namely, in a sample of N particles obtained through the corresponding algorithms, the distributions of the particles are correlated -although any finite number of them is asymptotically independent with respect to the total number N of particles. In the present article, exploiting the fine asymptotics of particle systems, we prove convergence theorems for U-statistics in this framework.

Published
2009

200. A Mean Field Theory of Nonlinear Filtering

Author

del Moral, Pierre, Patras, Frédéric, Rubenthaler, Sylvain, 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), 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), Laboratoire Jean Alexandre Dieudonné (JAD), Université Nice Sophia Antipolis (... - 2019) (UNS), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS), INRIA, 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), and Université Nice Sophia Antipolis (1965 - 2019) (UNS)

Subjects
Feynman-Kac measures, interacting particle systems, propagations of chaos, [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], combinatorial enumeration, nonlinear filtering, historical and genealogical tree models, trees and forests, 60G35, 47D08, 60C05, 60K35, 65C35, 65D30, 31B10, 60J80, 60F17, 65C05, 92D25, central limit theorems, Gaussian fields
Abstract

We present a mean field particle theory for the numerical approximation of Feynman-Kac path integrals in the context of nonlinear filtering. We show that the conditional distribution of the signal paths given a series of noisy and partial observation data is approximated by the occupation measure of a genealogical tree model associated with mean field interacting particle model. The complete historical model converges to the McKean distribution of the paths of a nonlinear Markov chain dictated by the mean field interpretation model. We review the stability properties and the asymptotic analysis of these interacting processes, including fluctuation theorems and large deviation principles. We also present an original Laurent type and algebraic tree-based integral representations of particle block distributions. These sharp and non asymptotic propagations of chaos properties seem to be the first result of this type for mean field and interacting particle systems.

Published
2008

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